Practising JEE Main mock tests is crucial for effective exam preparation. These tests help students understand the exam pattern, improve time management, and identify strengths and weaknesses. Regular practice boosts confidence, enhances problem-solving speed, and ensures better performance in the actual exam.

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JEE Main Application of Derivatives Practice Questions with Solutions

Application of Derivatives is an important topic from the Mathematics section of the JEE Main exam. It is also a scoring section, so students should make the most out of it through regular practice. The JEE Main Application of Derivatives Practice Test is a valuable resource for all JEE Main aspirants to learn about important mathematical concepts and enhance their problem-solving abilities. The JEE Main Application of Derivatives Practice Questions with Solutions have many different questions from the Sequences and Series chapter for students to practice and strengthen their mathematics section for the JEE Main entrance exam.

About 2 to 3 questions may appear from this chapter on the JEE Main question paper. You will get multiple-choice questions and integer-type questions in the question paper, which will test your problem-solving abilities. JEE Main is a challenging entrance exam for all engineering aspirants. Therefore, it is advised for candidates to attempt these practice tests at least once a week to develop knowledge and precision of the topic.

Around 15 to 20 questions are available from this chapter in the JEE Main Application of Derivatives Practice Questions with Solutions, covering all the essential topics. The important topics of this chapter, from which questions are likely to appear in the exam, are Motion in a Straight Line, Monotonicity, Rate of Change of Quantities, Maxima and Minima, Tangents and Normals, Analysis of Graphs & Curvature, and Rolle’s & Lagrange’s Mean Value Theorems. The practice tests will contain questions on all these topics, along with their detailed solutions for students to practice. Students may go through the solutions and their explanations provided for their understanding.

Practising the JEE Main Application of Derivatives Practice Questions with Solutions on a regular basis will help the students to improve their accuracy in solving problems and manage their time better. It will also allow them to recall formulas, understand concepts, and solve questions from this topic easily in the JEE Main exam.

JEE Main Mathematics Application of Derivatives Practice Questions

Chemistry Physics

Question 1.

If the function $f (x) = 2 x^{3} - 9 {ax}^{2} + 12 a^{2} x + 1, a > 0$ has a local maximum at $x = α$ and a local minimum at $x = α^{2}$ , then $α$ and $α^{2}$ are the roots of the equation :

x^{2} - 6 x + 8 = 0

8 x^{2} - 6 x + 1 = 0

8 x^{2} + 6 x - 1 = 0

x^{2} + 6 x + 8 = 0

Question 2.

Let $f (x) = 4 \cos^{3} x + 3 \sqrt{3} \cos^{2} x - 10$ . The number of points of local maxima of $f$ in interval $(0, 2 π)$ is

Question 3.

The number of critical points of the function $f (x) = (x - 2)^{2 / 3} (2 x + 1)$ is

Question 4.

For the function $f (x) = (\cos x) - x + 1, x \in R$ , between the following two statements

(S1) $f (x) = 0$ for only one value of $x$ in $[0, π]$ .

(S2) $f (x)$ is decreasing in $[0, \frac{π}{2}]$ and increasing in $[\frac{π}{2}, π]$ .

Both (S1) and (S2) are incorrect.

Only (S1) is correct.

Only (S2) is correct.

Both (S1) and (S2) are correct.

Question 5.

The interval in which the function $f (x) = x^{x}, x > 0$ , is strictly increasing is

(0, \infty)

(0, \frac{1}{e}]

[\frac{1}{e^{2}}, 1)

[\frac{1}{e}, \infty)

Question 6.

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a+b) $^{2}$ is equal to :

Question 7.

Let $f (x) = x^{5} + 2 x^{3} + 3 x + 1, x \in R$ , and $g (x)$ be a function such that $g (f (x)) = x$ for all $x \in R$ . Then $\frac{g (7)}{g^{'} (7)}$ is equal to :

Question 8.

For the function

$f (x) = \sin x + 3 x - \frac{2}{π} (x^{2} + x), where x \in [0, \frac{π}{2}],$

consider the following two statements :

(I) $f$ is increasing in $(0, \frac{π}{2})$ .

(II) $f^{'}$ is decreasing in $(0, \frac{π}{2})$ .

Between the above two statements,

only (I) is true.

both (I) and (II) are true.

only (II) is true.

neither (I) nor (II) is true.

Question 9.

Let $f (x) = 3 \sqrt{x - 2} + \sqrt{4 - x}$ be a real valued function. If $α$ and $β$ are respectively the minimum and the maximum values of $f$ , then $α^{2} + 2 β^{2}$ is equal to

Question 10.

Let the sum of the maximum and the minimum values of the function $f (x) = \frac{2 x^{2} - 3 x + 8}{2 x^{2} + 3 x + 8}$ be $\frac{m}{n}$ , where $\gcd (m, n) = 1$ . Then $m + n$ is equal to :

217

182

201

195

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Question 1.

5 f (x) + 4 f (\frac{1}{x}) = x^{2} - 2, \forall x \neq 0

and

y = 9 x^{2} f (x)

, then

y

is strictly increasing in :

(0, \frac{1}{\sqrt{5}}) \cup (\frac{1}{\sqrt{5}}, \infty)

(- \frac{1}{\sqrt{5}}, 0) \cup (\frac{1}{\sqrt{5}}, \infty)

(- \frac{1}{\sqrt{5}}, 0) \cup (0, \frac{1}{\sqrt{5}})

(- \infty, \frac{1}{\sqrt{5}}) \cup (0, \frac{1}{\sqrt{5}})

Question 2.

Let $f :\to R \to (0, \infty)$ be strictly increasing function such that $lim_{x \to \infty} \frac{f (7 x)}{f (x)} = 1$ . Then, the value of $lim_{x \to \infty} [\frac{f (5 x)}{f (x)} - 1]$ is equal to

7/5

Question 3.

If the function $f : (- \infty, - 1] \to (a, b]$ defined by $f (x) = e^{x^{3} - 3 x + 1}$ is one - one and onto, then the distance of the point $P (2 b + 4, a + 2)$ from the line $x + e^{- 3} y = 4$ is :

2 \sqrt{1 + e^{6}}

\sqrt{1 + e^{6}}

3 \sqrt{1 + e^{6}}

4 \sqrt{1 + e^{6}}

Question 4.

$If f (x) = | \begin{array}{ccc} x^{3} & 2 x^{2} + 1 & 1 + 3 x \\ 3 x^{2} + 2 & 2 x & x^{3} + 6 \\ x^{3} - x & 4 & x^{2} - 2 \end{array} | for all x \in R, then 2 f (0) + f^{'} (0) is equal to$

Question 5.

Let $f (x) = (x + 3)^{2} (x - 2)^{3}, x \in [- 4, 4]$ . If $M$ and $m$ are the maximum and minimum values of $f$ , respectively in $[- 4, 4]$ , then the value of $M - m$ is

108

392

608

600

Question 6.

The maximum area of a triangle whose one vertex is at $(0, 0)$ and the other two vertices lie on the curve $y = - 2 x^{2} + 54$ at points $(x, y)$ and $(- x, y)$ , where $y > 0$ , is :

108

122

Question 7.

The function $f (x) = \frac{x}{x^{2} - 6 x - 16}, x \in R - {- 2, 8}$

decreases in

(- \infty, - 2) \cup (- 2, 8) \cup (8, \infty)

increases in

(- \infty, - 2) \cup (- 2, 8) \cup (8, \infty)

decreases in

(- 2, 8)

and increases in

(- \infty, - 2) \cup (8, \infty)

decreases in

(- \infty, - 2)

and increases in

(8, \infty)

Question 8.

The function $f (x) = 2 x + 3 (x)^{\frac{2}{3}}, x \in R$ , has

exactly one point of local minima and no point of local maxima

exactly one point of local maxima and exactly one point of local minima

exactly two points of local maxima and exactly one point of local minima

exactly one point of local maxima and no point of local minima

Question 9.

Consider the function $f : [\frac{1}{2}, 1] \to R$ defined by $f (x) = 4 \sqrt{2} x^{3} - 3 \sqrt{2} x - 1$ . Consider the statements

(I) The curve $y = f (x)$ intersects the $x$ -axis exactly at one point.

(II) The curve $y = f (x)$ intersects the $x$ -axis at $x = \cos \frac{π}{12}$ .

Then

Both (I) and (II) are correct.

Only (I) is correct.

Both (I) and (II) are incorrect.

Only (II) is correct.

Question 10.

Let $g (x) = 3 f (\frac{x}{3}) + f (3 - x)$ and $f^{''} (x) > 0$ for all $x \in (0, 3)$ . If $g$ is decreasing in $(0, α)$ and increasing in $(α, 3)$ , then $8 α$ is :

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JEE Main Application of Derivatives Practice Questions With Solutions