Calculus

Application of Derivatives

Medium

core

The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)

decreasing on $\mathbb{R}$

increasing on $\mathbb{R}$

increasing on $(0, \infty)$

neither increasing nor decreasing on $(-\infty, 0)$

✅ Correct Option: 1

To check if a function is increasing or decreasing, find its derivative and check if it's positive or negative.

If $f'(x) > 0$ , the function is increasing.

If $f'(x) < 0$ , the function is decreasing.

Given: $f(x) = 4 - 3x + 3x^2 - x^3$

Using the power rule:

$f'(x) = 0 - 3(1) + 3(2x) - 3x^2$

$f'(x) = -3 + 6x - 3x^2$

Factoring out the common factor $-3$ :

$f'(x) = -3(1 - 2x + x^2)$

The expression $1 - 2x + x^2$ is a perfect square since $(x - 1)^2 = x^2 - 2x + 1$

Therefore:

$f'(x) = -3(x - 1)^2$

The term $(x - 1)^2$ is always $\geq 0$ for any value of $x$ .

Therefore:

$-3(x - 1)^2 \leq 0$ for all $x$

This means $f'(x) \leq 0$ for all real numbers.

Since $f'(x) \leq 0$ for all $x \in \mathbb{R}$ , the function is decreasing on $\mathbb{R}$ .

Note: Although $f'(x) = 0$ at $x = 1$ , this is just one point. The function is still considered decreasing on all of $\mathbb{R}$ .

Therefore, the function is decreasing on $\mathbb{R}$ .

2026: 15 May Shift 1

Match the LIST-I with LIST-II

	LIST-I (Function)		LIST-II (Critical point)
A.	$y = x^2 + x + 1$	I.	$2/3$
B.	$y = 2x^3 - 6x^2 + 6x$	II.	$-1$
C.	$y = 2x^2 + 4x - 1$	III.	$-1/2$
D.	$y = x^3 - x^2$	IV.	$1$

Choose the correct answer from the options given below:

2026: 18 May Shift 2

For the function $f(x)=2x^3-21x^2+36x-100$ ;

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. Point of local maxima is	I. -208
B. Point of local minima is	II. -83
C. Local maximum value is	III. 6
D. Local minimum value is	IV. 1

Choose the correct answer from the options given below:

2026: 25 May Shift 1

The points in the interval $[0,2\pi]$ , where the function $f(x) = \cos2x + 2\sin x\cos x$ , attains its local minima, is/are:

A. $\pi/8$

B. $5\pi/8$

C. $13\pi/8$

D. $11\pi/8$

Choose the correct answer from the options given below:

2025: 30 May Shift 2

The function $f: R \rightarrow R$ (where $R$ is set of real numbers) defined as

$f(x) = x^2 + 2x$ is

2026: 31 May Shift 2

The function $f(x) = 24x^4 - 32x^3 - 96x^2 + 7$ on the interval $\left[-\dfrac{3}{2}, 4\right]$ is :

A. increasing in $(-1,0)$

B. decreasing in $(0,4)$

C. increasing in $(2,4)$

D. decreasing in $\left(-\dfrac{3}{2},-1\right)$

Choose the correct answer from the options given below:

2025: 26 May Shift 2

If it is given that at $x = 1$ , the function $f(x) = x^4 - 62x^2 + 2ax + b$ attains its maximum value on the interval [0, 2], then the value of a is:

The function f(x)=4−3x+3x2−x3f(x) = 4 - 3x + 3x^2 - x^3f(x)=4−3x+3x2−x3 is (Here R\mathbb{R}R is set of real numbers)

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)