Calculus

Application of Derivatives

Medium

common

The function $f(x) = x^3 + 3x^2 + 4x + 4$ , $x \in \mathbb{R}$ (set of real numbers) :

is increasing on $\mathbb{R}$

is decreasing on $\mathbb{R}$

is decreasing on $(-\infty, 0)$

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

✅ Correct Option: 1

To determine if the function is increasing or decreasing, the derivative $f'(x)$ is examined. If $f'(x) > 0$ for all $x$ , the function is increasing. If $f'(x) < 0$ for all $x$ , the function is decreasing.

$f(x) = x^3 + 3x^2 + 4x + 4$

Using the power rule:

$f'(x) = 3x^2 + 6x + 4$

To determine the sign of $f'(x) = 3x^2 + 6x + 4$ , the discriminant is calculated.

For the quadratic $ax^2 + bx + c$ where $a = 3$ , $b = 6$ , $c = 4$ :

$D = b^2 - 4ac$

$D = (6)^2 - 4(3)(4)$

$D = 36 - 48$

$D = -12$

Since $D < 0$ , the quadratic $f'(x) = 3x^2 + 6x + 4$ has no real roots.

Since the coefficient of $x^2$ is positive ( $a = 3 > 0$ ), the parabola opens upward.

Therefore, $f'(x) > 0$ for all $x \in \mathbb{R}$ .

Since $f'(x) > 0$ for all $x \in \mathbb{R}$ , the function $f(x) = x^3 + 3x^2 + 4x + 4$ is increasing on $\mathbb{R}$ .

2026: 19 May Shift 1

For the function $f(x) = \frac{x^4}{4} - 2x^3 + \frac{11}{2}x^2 - 6x$ , which of the following statements are TRUE?

(A) Critical points of $f(x)$ are $x = 1, 2$ and $3$ .

(B) $f(x)$ is increasing over $(1,2)$ .

(D) $f(x)$ is increasing over $(3,\infty)$ .

Choose the correct answer from the options given below:

2026: 22 May Shift 1

Sand is pouring from a pipe at the rate of 12 $cm^3$ /sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm ?

2026: 21 May Shift 1

The points of local maxima and local minima of the $f(x) = x^3 - 6x^2 + 9x + 27$ are

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 ____

2026: 21 May Shift 1

Match List-I with List-II

List-I	List-II
(A) The interval in which $f(x) = x^2 - 4x + 17$ is increasing.	(I) $(-5,\infty)$
(B) The interval in which $f(x) = x^2 - 4x + 17$ is decreasing.	(II) $(-\infty,-5)$
(C) The interval in which $f(x) = 3x^2 + 30x - 7$ is increasing.	(III) $(2,\infty)$
(D) The interval in which $f(x) = 3x^2 + 30x - 7$ is decreasing.	(IV) $(-\infty,2)$

Choose the correct answer from the options given below:

2025: 14 May Shift 1

If the maximum value of the function $f(x) = \frac{\log_ex}{x}$ , $x > 0$ occurs at $x = a$ , then $a^2f''(a)$ is equal to

The function f(x)=x3+3x2+4x+4f(x) = x^3 + 3x^2 + 4x + 4f(x)=x3+3x2+4x+4, x∈Rx \in \mathbb{R}x∈R (set of real numbers) :

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The function $f(x) = x^3 + 3x^2 + 4x + 4$ , $x \in \mathbb{R}$ (set of real numbers) :