Calculus

Application of Derivatives

Medium

common

The function $f(x) = x^4 - 2x^2$ is increasing on

$(-1, 0) \cup (1, \infty)$

$(-\infty, -1) \cup (0, 1)$

$(-\infty, \infty)$

$(-\infty, 0) \cup (1, \infty)$

✅ Correct Option: 1

A function is increasing when the derivative is positive: $f'(x) > 0$ .

Given $f(x) = x^4 - 2x^2$

Using the power rule:

$f'(x) = 4x^3 - 4x$

Factoring:

$f'(x) = 4x(x^2 - 1)$

$f'(x) = 4x(x - 1)(x + 1)$

Setting $f'(x) = 0$ :

$4x(x - 1)(x + 1) = 0$

Critical points: $x = -1, x = 0, x = 1$

These points divide the number line into 4 intervals.

Testing the sign of $f'(x) = 4x(x - 1)(x + 1)$ in each interval:

Interval	Test Point	Sign of 4x	Sign of (x-1)	Sign of (x+1)	Sign of f'(x)	Increasing?
$x < -1$	$x = -2$	-	-	-	-	No
$-1 < x < 0$	$x = -0.5$	-	-	+	+	Yes
$0 < x < 1$	$x = 0.5$	+	-	+	-	No
$x > 1$	$x = 2$	+	+	+	+	Yes

The derivative is positive on $(-1, 0)$ and $(1, \infty)$ .

The function is increasing on $(-1, 0) \cup (1, \infty)$ .

2025: 13 May Shift 1

In which of the following interval the function $f(x) = x^x, x > 0$ is strictly increasing?

2025: 2 June Shift 1

If maximum value of $f(x) = 2x^3 + 3x^2 - 6ax + 10$ occurs at $x = -3$ , then the value of $\alpha$ is ____

2023: 22 May Shift 3

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to :

2026: 22 May Shift 1

An angle $\theta, 0<\theta<\frac{\pi}{2}$ , which increases twice as fast as its sine, is

2026: 30 May Shift 2

For which of the following real functions, $x=0$ is a point of local maxima?

(A) $f(x)=4x^4-3x^3-2x^2+1$

(B) $f(x)=3x^5+\sqrt{2}x^4-7x^2+4$

(D) $f(x)=-3x^3+2x^2-9$

Choose the correct answer from the options given below:

2026: 12 May Shift 2

A function $f(x)$ is defined as:

$f(x) = x^3-6x^2+9x+15$

For the function $f(x)$ , which of the following statements are correct ?

(A) The critical points are $x=1$ and $x=3$ .

(B) $x=1$ is a point of local minimum.

(D) $x=3$ is a point of local minimum.

Choose the correct answer from the options given below:

The function f(x)=x4−2x2f(x) = x^4 - 2x^2f(x)=x4−2x2 is increasing on

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The function $f(x) = x^4 - 2x^2$ is increasing on