Application of Derivatives27 questions4 PYQ

Application of Derivatives — JEE Maths Practice Questions & Solutions

27 questions on Application of Derivatives with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

mediumPYQ · JEE Main 2026

For the functions

f(\theta)=\alpha\tan^2\theta+\beta\cot^2\theta

and

g(\theta)=\alpha\sin^2\theta+\beta\cos^2\theta

\alpha>\beta>0

, let

\displaystyle\min_{0<\theta<\pi/2}f(\theta)=\max_{0<\theta<\pi}g(\theta)

. If the first term of a G.P. is

\left(\dfrac{\alpha}{2\beta}\right)

, its common ratio is

\left(\dfrac{2\beta}{\alpha}\right)

, and the sum of its first

10

terms is

\dfrac{m}{n}

with

\gcd(m,n)=1

, then

m+n

is equal to

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mediumPYQ · JEE Main 2026

Let

f:\mathbb{R}\to\mathbb{R}

be a differentiable function such that

f\left(\dfrac{x+y}{3}\right)=\dfrac{f(x)+f(y)}{3}

for all

x,y\in\mathbb{R}

and

f'(0)=3

. Then the minimum value of

g(x)=3+e^x f(x)

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mediumPYQ · JEE Main 2026

\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

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mediumPYQ · JEE Main 2026

The number of critical points of the function

f(x)=\begin{cases}\left|\dfrac{\sin x}{x}\right|, & x\ne0\\[4pt] 1, & x=0\end{cases}

in the interval

(-2\pi,2\pi)

is equal to

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hard

Consider the function, for

x \in [-2, 3]

f(x) = \begin{cases} \dfrac{x^{3}-2x^{2}-5x+6}{x-1} & \text{if } x \ne 1 \\[2mm] -6 & \text{if } x = 1 \end{cases}

f(x) = \displaystyle\lim_{m\to\infty}\frac{\ln\left(3+x^2\right)-x^{2m}\sin\left(x^2\right)}{1+x^{2m}}

, then which of the following is/are CORRECT?

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hard

x

and

y

are real numbers such that

x^{2}+y^{2} = 27

, then the maximum possible value of

x-y

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hard

The function

f(x) = \frac{x^{2}-2}{\sqrt{1+x^{2}}}

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hard

Let

f(x) = \int_{0}^{x^{2}} (t-1)(t-4)(t-9)\,dt ,

f

satisfies

f\left(x\right)\left(f\left(y\right)\right)^{-1} \le 2^{\left(x-y\right)^2}

\forall\,x,y \in D_f

, where

D_f

is the domain set of

f

. Then

f\left(x\right)

f : \mathbb{R} \to \mathbb{R}

satisfy the functional equation

f(x+y) = f(x)\cdot f(y),

where

f(0) \ne 0

and

f'(0) = 1

. Given

g(x) = \frac{3x}{f(x)},

f(x)

be a derivable function which is increasing for all

x \in \mathbb{R}

(having no critical point), then

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medium

The set of values of

a

for which the function

f(x) = (a+2)x^{3} - 3ax^{2} + 9ax - 1

is decreasing for all real

x

, is

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medium

f(x) = \begin{cases} 2x+3 & x < 0 \\ 5 & x = 0 \\ x^{2}+7 & x > 0 \end{cases}

then at

x = 0

f(x)

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medium

The complete set of values of

m

, for which the function

f(x) = 3 + mx + e^{-x}

is always decreasing, is

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medium

If the points of local extremum of

f(x) = x^{3}-3ax^{2}+3\left(a^{2}-1\right)x+1

lie between

-2

and

4

, then

a

belongs to

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medium

x \cdot y = 1

, then the minimum value of

x^{2}+y^{2}

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medium

Let

f(x)

be a function defined by

f(x) = \int_{3}^{x} t\left(t^{2}-7t+12\right)dt , \qquad 3 \le x \le 6 .

If the range of

f(x)

[a,\ b]

, then the value of \left(3a + \dfrac {9}\right) is

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medium

Given

f(x) = \begin{cases} |x-1|+5 & \text{if } x \ne 1 \\ 8 & \text{if } x = 1 \end{cases}

then at

x = 1

f(x)

has

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medium

The number of distinct real roots of the equation

2x^{4}-8x^{3}+8x^{2}-1 = 0

n

, then the value of

\dfrac1n

f

be a function which is continuous and differentiable for all real

x

. If

f(2) = -4

and

f'(x) \ge 6

for all

x \in [2,4]

f

g

be two continuous and twice derivable functions such that

f(0) = f(3) = 0

;

f(1)\cdot f(2) < 0

;

g(0) = g(3) = 0

. The number of roots of the equation

f''(x)\,g(x) + f'(x)\,g'(x) = 0

(-1, 5)

cannot be

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medium

Water is flowing into a container at a rate proportional to the square of the amount of water present inside the container. If at

t = 0

the volume of water inside the tank is

V_0

, and it takes

6

hours for the volume to become

2V_0

, then the time (in hours) taken for the volume to become

4V_0

is equal to

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medium

f(x) = 3x^{2} + \int_{0}^{x} e^{-t} f(x-t)\,dt ,

a

for which

a^x = \log_ax

has exactly

3

solutions is

\left(0,\ b^{-b}\right)

, then

\left[b\right]

is, where

\left[\ \cdot\ \right]

is G.I.F.

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easy

The number of non-negative integers which satisfy the inequation

f\left(f\left(f(x)\right)\right) > f\left(f(-x)\right),

where

f(x) = 30 - 2x - x^{3}

f

be a differentiable function for all

x \in \mathbb{R}

, where

f(1) = 1

and

f(3) = 4

and

f'(x) \ge 1

for all

x \in \mathbb{R}

. Then

f(2)

can be equal to

View solution →

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