JEE Main Application of Derivatives Practice Questions With Solutions

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JEE Main Mathematics Application of Derivatives Practice Questions

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

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

JEE Main Mathematics Application of Derivatives Practice Questions

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

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