Applications of Derivatives

9 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 2·Q2·MCQ·Moderate
Open
Monotonicity of composite functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbb{R}$ and $f'(a-1) = 0$ , where $a$ is real number. Let $g(x) = f(\tan^2 x - 2\tan x + a)$ , $0 < x < \dfrac{\pi}{2}$ .
Consider the following two statements:
(I) $g$ is increasing in $\left(0, \dfrac{\pi}{4}\right)$
(II) $g$ is decreasing in $\left(\dfrac{\pi}{4}, \dfrac{\pi}{2}\right)$
Then,
1. (1)Neither (I) nor (II) is True
2. (2)Only (II) is True
3. (3)Only (I) is True
4. (4)Both (I) and (II) are True
MathsJEE Main 22 Jan 2026, Shift 1·Q5·MCQ·Easy–moderate
Open
Absolute minimum on an interval
Let $f(x) = x^{2025} - x^{2000}$ , $x \in [0, 1]$ and the minimum value of the function $f(x)$ in the interval $[0, 1]$ be $(80)^{80}(n)^{-81}$ . Then $n$ is equal to
1. (1) $-81$
2. (2) $-40$
3. (3) $-41$
4. (4) $-80$
MathsJEE Main 24 Jan 2026, Shift 1·Q24·Numerical·Moderate
Open
Monotonicity and local maxima
Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \dfrac{\lvert t + 1\rvert}{t^2}$ , $t < 0$ , is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x - 2) + \alpha x^2 + 4x - \alpha$ , $x > 2$ , is .
MathsJEE Main 24 Jan 2026, Shift 2·Q15·MCQ·Moderate
Open
Monotonicity of functions
Consider the following three statements for the function $f : (0, \infty) \to \mathbb{R}$ defined by $f(x) = \lvert\log_e x\rvert - \lvert x - 1\rvert$ :
(I) $f$ is differentiable at all $x > 0$ .
(II) $f$ is increasing in $(0, 1)$ .
(III) $f$ is decreasing in $(1, \infty)$ .
Then.
1. (1)All (I), (II) and (III) are TRUE.
2. (2)Only (I) is TRUE.
3. (3)Only (II) and (III) are TRUE.
4. (4)Only (I) and (III) are TRUE.
MathsJEE Main 28 Jan 2026, Shift 2·Q23·Numerical·Tough
Open
Local maxima and minima
Let $f$ be a differentiable function satisfying $f(x) = 1 - 2x + \int_0^x e^{(x-t)}f(t)\,dt,\ x \in \mathbb{R}$ and let $g(x) = \int_0^x (f(t) + 2)^{15}(t - 4)^6(t + 12)^{17}\,dt,\ x \in \mathbb{R}.$ If $p$ and $q$ are respectively the points of local minima and local maxima of $g$ , then the value of $\lvert p + q\rvert$ is equal to .
MathsJEE Main 2 Apr 2026, Shift 1·Q18·MCQ·Tough
Open
Critical points of a function
The number of critical points of the function $f(x) = \begin{cases} \left\lvert\dfrac{\sin x}{x}\right\rvert, & x \neq 0\\[6pt] 1, & x = 0\end{cases}$ in the interval $(-2\pi, 2\pi)$ is equal to:
1. (1)1
2. (2)3
3. (3)5
4. (4)7
MathsJEE Main 2 Apr 2026, Shift 2·Q18·MCQ·Moderate
Open
Polynomial with given extrema
Let $f(x)$ be a polynomial of degree 5, and have extrema at $x = 1$ and $x = -1$ . If $\displaystyle\lim_{x\to0}\left(\frac{f(x)}{x^3}\right) = -5$ , then $f(2) - f(-2)$ is equal to:
1. (1)0
2. (2)50
3. (3)92
4. (4)112
MathsJEE Main 4 Apr 2026, Shift 2·Q12·MCQ·Moderate
Open
Maxima and minima
$\displaystyle\max_{0 \le x \le \pi}\left(16\sin\left(\frac{x}{2}\right)\cos^3\left(\frac{x}{2}\right)\right)$ is equal to:
1. (1) $\dfrac{3\sqrt3}{2}$
2. (2) $3\sqrt3$
3. (3) $4\sqrt3$
4. (4) $6\sqrt3$
MathsJEE Main 6 Apr 2026, Shift 2·Q17·MCQ·Moderate
Open
Maximum and minimum on interval
Let $A = \begin{bmatrix} 1 & 3 & -1\\ 2 & 1 & \alpha\\ 0 & 1 & -1\end{bmatrix}$ be a singular matrix. Let $f(x) = \displaystyle\int_0^x \left(t^2 + 2t + 3\right)dt$ , $x \in [1, \alpha]$ . If $M$ and $m$ are respectively the maximum and the minimum values of $f$ in $[1, \alpha]$ then $3(M - m)$ is equal to:
1. (1)64
2. (2)68
3. (3)72
4. (4)76
We corrected a printing error in the original paper
Source printed the matrix $A$ between determinant bars (“ $A = \lvert\ldots\rvert$ be a singular matrix”).