2. Calculus — Differential Calculus

Decreasing functions

Q: FAQ 1: Is a function decreasing if $f'(x) = 0$ at a single isolated point?

Yes. For example, $f(x) = -x^3$ has $f'(0) = 0$, but it is strictly decreasing across the entire real line because $f(x_1) > f(x_2)$ whenever $x_1 < x_2$. Zero derivative at isolated points does not destroy strict monotonicity.

Q: FAQ 2: Why do competitive exam questions frequently test quadratic derivative discriminants?

Examiners test whether candidates understand the global sign requirement of a quadratic ($A x^2 + B x + C \le 0$ for all $x$). This requires both $A < 0$ and $B^2 - 4AC \le 0$, which quickly filters out unprepared aspirants.

Q: FAQ 3: How can I save time during the exam when finding intervals of decrease?

For cubic polynomials, immediately find the roots of $f'(x) = 0$. If the leading coefficient is positive, the middle interval between the two roots is the decreasing region. Use this shortcut to mark answers.

A real-valued function is said to be decreasing on an interval if its graph slopes downward or remains level from left to right, characterized analytically by a non-positive first derivative ($f'(x) \le 0$).

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Monotonicity Derivative Test (Decreasing Function Criterion)

Let $f: [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. \n1. If $f'(x) < 0$ for all $x \in (a, b)$, then $f$ is strictly decreasing on $[a, b]$. \n2. If $f'(x) \le 0$ for all $x \in (a, b)$, then $f$ is decreasing (non-increasing) on $[a, b].$

📐 Geometric Interpretation

Geometrically, the tangent line drawn at any point on the curve of a decreasing function has a non-positive slope ($m = \tan \theta \le 0$, where $\theta \in [\pi/2, \pi)$), implying that as the independent variable $x$ increases along the horizontal axis, the dependent variable $y = f(x)$ moves downward or stays flat along the vertical axis.

⚙️ Algebraic & Structural Meaning

Algebraically, for any two points $x_1, x_2$ in the interval such that $x_1 < x_2$, a decreasing function satisfies $f(x_1) \ge f(x_2)$ (and $f(x_1) > f(x_2)$ for strictly decreasing functions), providing an inequality tool to bound algebraic and transcendental expressions without plotting.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Continuity on the closed interval	`[a, b] (Closed interval)`	Strictly Mandatory / Necessary	If $f$ is discontinuous at any interior point, the global monotonicity property between endpoints breaks down, and intermediate values can jump arbitrarily.
Differentiability on the open interval	`(a, b) (Open interval)`	Strictly Mandatory	If derivative fails to exist (e.g., sharp corners or cusps), standard derivative tests for monotonicity are invalid and must be replaced by direct inequality testing.
Sign of First Derivative	`(a, b) (Open interval)`	Essential for conclusion	If $f'(x)$ changes sign (both positive and negative values in $(a, b)$), the function is neither increasing nor decreasing on the entire interval, but rather oscillatory or piecewise monotonic.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Polynomial Derivative Sign Test

$$f'(x) = a_n x^{n-1} + \dots + a_1 \le 0$$

Rule: For quadratic polynomials $f(x) = ax^2 + bx + c$ with $a > 0$, the function is never decreasing on $\mathbb{R}$. For cubic polynomials $f(x) = ax^3 + bx^2 + cx + d$, find roots of $f'(x) = 0$ as $\alpha, \beta$ ($\alpha < \beta$); the function decreases on $[\alpha, \beta]$ if leading coefficient $a > 0$ and $b^2 - 3ac > 0$.

Application: Directly tested in UP PGT to find the interval where a cubic or quartic polynomial is decreasing.

Trigonometric Monotonicity Intervals

$$f(x) = \sin x \implies f'(x) = \cos x \le 0 \text{ on } \left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$$

Rule: Remember standard reductions: $\sin x$ decreases on $[\pi/2, 3\pi/2]$, $\cos x$ decreases on $[0, \pi]$, and $\tan x$ is strictly increasing wherever continuous (never decreasing on any domain interval).

Application: Frequently asked in TGT/PGT matching lists and range-bound questions.

Exponential and Logarithmic Monotonicity

$$f(x) = a^x \text{ is decreasing if } 0 < a < 1, \quad f(x) = \log_a x \text{ is decreasing if } 0 < a < 1$$

Rule: Exponential and logarithmic functions reverse inequality when the base $a$ lies between $0$ and $1$. Use $a^{-x} = (1/a)^x$ to instantly spot decreasing exponentials.

Rational Function Sign Rule

$$f(x) = \frac{P(x)}{Q(x)} \implies f'(x) = \frac{P'(x)Q(x) - P(x)Q'(x)}{[Q(x)]^2} \le 0$$

Rule: Since the denominator $[Q(x)]^2$ is always positive where defined, ignore it and analyze only the sign of the numerator polynomial $P'(x)Q(x) - P(x)Q'(x)$ using Wavy Curve Method.

Composite Function Monotonicity Law

$h(x) = f(g(x)) \implies h'(x) = f'(g(x)) \cdot g'(x)$

Rule: If $f$ and $g$ have the same monotonicity (both increasing or both decreasing), their composite $f \circ g$ is increasing. If one is increasing and the other is decreasing, the composite is decreasing.

Parameter Range via Discriminant

$f'(x) = A x^2 + B x + C \le 0 \quad \forall x \in \mathbb{R} \implies A < 0 \text{ and } B^2 - 4AC \le 0$

Rule: To find values of parameter $k$ for which a quadratic derivative is non-positive for all real $x$, enforce the parabola opening downward ($A < 0$) and having at most one real root ($D \le 0$).

Application: Extremely common in UP PGT 5-mark and 4-mark short questions.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Strict vs Non-Strict Inequality Confusion

Common Pitfall: Candidates use $f'(x) < 0$ strictly when the question asks for 'decreasing' (which allows $f'(x) = 0$ at isolated points, such as $f(x) = -x^3$ at $x=0$).

Counterexample: $f(x) = -x^3 \implies f'(0) = 0$, yet $f(-1) = 1 > f(1) = -1$ (strictly decreasing on $\mathbb{R}$)

Why it fails: A function can be strictly decreasing even if its derivative vanishes at isolated points where the tangent is horizontal.

🛡️ Exam Defense: Check if equality $f'(x) = 0$ holds only at isolated points; if so, the function remains strictly decreasing on the entire interval.

⚠️ COMMON ERROR #02

Disregarding Domain Restrictions in Rational Functions

Common Pitfall: Including points where the denominator is zero inside the stated decreasing interval.

Counterexample: $f(x) = \frac{1}{x} \implies f'(x) = -\frac{1}{x^2} < 0$

Why it fails: While $f'(x) < 0$ for all $x \neq 0$, the function is NOT decreasing on $(-\infty, \infty)$ because $f(-1) = -1$ is not greater than $f(1) = 1$ across the vertical asymptote at $x = 0$.

🛡️ Exam Defense: Always exclude points of discontinuity and vertical asymptotes when stating intervals of monotonicity.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Strictly Decreasing	Continuous on $[a,b]$, Differentiable on $(a,b)$	$f'(x) < 0$ on $(a,b)$	Tangent slopes are strictly negative; strictly downward curve	Words: 'strictly decreasing', 'one-to-one'
Decreasing (Non-increasing)	Continuous on $[a,b]$, Differentiable on $(a,b)$	$f'(x) \le 0$ on $(a,b)$	Curve goes down or stays flat; no upward segments	Words: 'decreasing', 'non-increasing'
Constant Function	Continuous on $[a,b]$, Differentiable on $(a,b)$	$f'(x) = 0$ on $(a,b)$	Horizontal line parallel to $x$-axis	Words: 'independent of $x$', 'constant'
Monotonicity via Wavy Curve	Polynomial or Rational $f'(x)$	Factorize $f'(x) = P(x)/Q(x)$	Alternating sign regions on real number line	Questions asking 'Find the interval in which...'

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Find the interval in which the function $f(x) = 2x^3 - 9x^2 + 12x + 1$ is decreasing.

(A) $[1, 2]$(B) $(-\infty, 1] \cup [2, \infty)$(C) $[0, 3]$(D) $(-\infty, 0]$

📚 Method A: Formal Textbook Derivation

Step 1: Compute derivative $f'(x) = 6x^2 - 18x + 12$. Step 2: Set $f'(x) < 0 \implies 6(x^2 - 3x + 2) < 0$. Step 3: Factorize $6(x-1)(x-2) < 0$. Step 4: By Wavy Curve Method, $x \in (1, 2)$. Thus, the function is decreasing on $[1, 2]$.

⚡ Method B: 25-Second Exam Shortcut

For $ax^3 + bx^2 + cx + d$ with $a > 0$, the function decreases between the roots of $f'(x)=0$. Roots of $x^2-3x+2=0$ are $1$ and $2$. Answer is $[1, 2]$.

💡 Tip: Look at the leading coefficient sign; if positive, the middle interval between roots is always the decreasing zone.

CANONICAL PROBLEM 02 Dual-Method Breakdown

The function $f(x) = \frac{\ln x}{x}$ is strictly decreasing on the interval:

(A) $(0, e)$(B) $(e, \infty)$(C) $(0, 1)$(D) $(1, e^2)$

📚 Method A: Formal Textbook Derivation

Step 1: Find $f'(x) = \frac{(1/x)x - \ln x(1)}{x^2} = \frac{1 - \ln x}{x^2}$. Step 2: For strictly decreasing, $f'(x) < 0 \implies \frac{1 - \ln x}{x^2} < 0$. Step 3: Since $x^2 > 0$ for $x \in (0, \infty)$, we need $1 - \ln x < 0 \implies \ln x > 1 \implies x > e$. Step 4: Hence, interval is $(e, \infty)$.

⚡ Method B: 25-Second Exam Shortcut

Evaluate numerator at standard points or recall logarithmic peak at $x = e$, after which derivative becomes negative.

💡 Tip: Remember that $\frac{\ln x}{x}$ increases on $(0, e]$ and decreases on $[e, \infty)$. This is a standard result frequently memorized by toppers.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: If $f(x) = x^3 - 3x^2 + 3x - 100$ is strictly decreasing for all $x$ in:

A $(-\infty, \infty)$

B empty set $\phi$

C $(0, 1)$

D $(-\infty, 1)$

Derivation & Analysis:

Here $f'(x) = 3x^2 - 6x + 3 = 3(x-1)^2$. Since $3(x-1)^2 \ge 0$ for all real $x$, $f'(x)$ is never strictly less than zero ($f'(x) < 0$ has no solution). Thus, the function is strictly increasing everywhere and never decreasing; hence the set of such $x$ is the empty set $\phi$.

MCQ 2: The interval for which $f(x) = \cos x$ is decreasing is:

A $(-\pi/2, \pi/2)$

B $(0, \pi)$

C $(\pi, 2\pi)$

D $(-\pi, 0)$

Derivation & Analysis:

We have $f'(x) = -\sin x$. For $f(x)$ to be decreasing, $-\sin x \le 0 \implies \sin x \ge 0$. In the standard interval $[0, \pi]$, $\sin x \ge 0$, so $\cos x$ is decreasing on $(0, \pi)$.

MCQ 3: The function $f(x) = x^9 + 3x^7 + 64$ is:

A strictly increasing on $\mathbb{R}$

B strictly decreasing on $\mathbb{R}$

C C) decreasing only for $x < 0$

D neither increasing nor decreasing

Derivation & Analysis:

Computing the derivative, $f'(x) = 9x^8 + 21x^6$. Since $x^8 \ge 0$ and $x^6 \ge 0$ for all $x \in \mathbb{R}$, $f'(x) > 0$ for all $x \neq 0$. Thus, $f$ is strictly increasing on $\mathbb{R}$.

MCQ 4: Values of $a$ for which $f(x) = \sin x - a x + b$ is decreasing on $\mathbb{R}$ are:

A $a < 1$

B $a \ge 1$

C $a \le -1$

D $a > 0$

Derivation & Analysis:

We require $f'(x) = \cos x - a \le 0$ for all $x \in \mathbb{R}$. This implies $\cos x \le a$ for all $x$. Since the maximum value of $\cos x$ is $1$, we must have $a \ge 1$.

MCQ 5: The function $f(x) = x^3 - 3ax^2 + 3ax + 1$ is strictly increasing for all $x \in \mathbb{R}$ if:

A $0 < a < 1$

B $a < 0$ or $a > 1$

C $a = 0$ or $a = 1$

D $0 \le a \le 1$

Derivation & Analysis:

Here $f'(x) = 3x^2 - 6ax + 3a$. For $f$ to be increasing everywhere, $f'(x) \ge 0$ for all $x \in \mathbb{R}$, which requires discriminant $B^2 - 4AC \le 0 \implies 36a^2 - 36a \le 0 \implies a(a-1) \le 0 \implies 0 \le a \le 1$.

MCQ 6: Which of the following functions is strictly decreasing on $(0, \pi/2)$?

A $\sin x$

B $\tan x$

C $\cos x$

D $\sec x$

Derivation & Analysis:

For $f(x) = \cos x$, $f'(x) = -\sin x$. On $(0, \pi/2)$, $\sin x > 0$, so $-\sin x < 0$, making $\cos x$ strictly decreasing. All other options have positive derivatives on $(0, \pi/2)$.

MCQ 7: If $f(x) = x e^{x(1-x)}$, then $f(x)$ is decreasing in the interval:

A $[-1/2, 1]$

B $(-\infty, -1/2] \cup [1, \infty)$

C $[0, 1]$

D $(-\infty, \infty)$

Derivation & Analysis:

Taking natural log and differentiating or using product rule gives $f'(x) = e^{x-x^2}(1 + x - 2x^2)$. Setting $f'(x) \le 0$ gives $2x^2 - x - 1 \ge 0 \implies (2x+1)(x-1) \ge 0$, which yields $x \in (-\infty, -1/2] \cup [1, \infty)$.

MCQ 8: Assertion (A): $f(x) = \frac{x}{\ln x}$ is decreasing on $(0, 1)$. Reason (R): Derivative of $f(x)$ is negative on $(0, 1)$.

A Both (A) and (R) are true and (R) is the correct explanation of (A)

B Both (A) and (R) are true but (R) is not the correct explanation

C (A) is true but (R) is false

D (A) is false but (R) is true

Derivation & Analysis:

Let's check $f'(x) = \frac{\ln x - 1}{(\ln x)^2}$. On $(0, 1)$, $\ln x < 0$, so numerator is negative and denominator is positive, making $f'(x) < 0$. However, $f(x)$ is NOT continuous/defined at $x=1$ and approaches $-\infty$ near $0$. Hence, Assertion (A) as stated on $(0, 1)$ fails at $x=1$ or requires careful domain handling, making R true but A false in standard strict formulation.

MCQ 9: The largest interval in which $f(x) = x^3 - 12x^2 + 36x + 17$ is decreasing is:

A $(2, 6)$

B $[2, 6]$

C $(-\infty, 2]$

D $[6, \infty)$

Derivation & Analysis:

$f'(x)=3x^2-24x+36=3(x-2)(x-6)$. The derivative is non-positive on $[2,6]$ and positive outside that interval, so the largest interval on which the function is decreasing is $[2,6]$.

Q10

MCQ 10: If $f(x) = kx^3 - 9x^2 + 9x + 3$ is strictly decreasing for all $x \in \mathbb{R}$, then:

A $k < 3$

B $k \le 3$

C $k \le 0$

D no such real value of $k$ exists

Derivation & Analysis:

Here $f'(x) = 3kx^2 - 18x + 9$. For this quadratic to be $\le 0$ for all $x \in \mathbb{R}$, we must have leading coefficient $3k < 0 \implies k < 0$, AND discriminant $B^2 - 4AC = 324 - 4(3k)(9) = 324 - 108k \le 0 \implies k \ge 3$. Since $k < 0$ and $k \ge 3$ cannot simultaneously hold, no such real value of $k$ exists.

Common questions

Yes. For example, $f(x) = -x^3$ has $f'(0) = 0$, but it is strictly decreasing across the entire real line because $f(x_1) > f(x_2)$ whenever $x_1 < x_2$. Zero derivative at isolated points does not destroy strict monotonicity.

Examiners test whether candidates understand the global sign requirement of a quadratic ($A x^2 + B x + C \le 0$ for all $x$). This requires both $A < 0$ and $B^2 - 4AC \le 0$, which quickly filters out unprepared aspirants.

For cubic polynomials, immediately find the roots of $f'(x) = 0$. If the leading coefficient is positive, the middle interval between the two roots is the decreasing region. Use this shortcut to mark answers.

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