The least value of the function $f(x)=2 \cos x+x$ in the closed interval $\left[0, \frac{\pi}{2}\right]$ is
- A2
- B$\frac{\pi}{6}+\sqrt{3}$
- C$\frac{\pi}{2}$
- DThe least value does not exist.
Question types
Application of Derivatives question types
262 questions across 7 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
262
Questions
7
Question groups
5
Question types
01
M.C.Q (1 Marks)
68 Q→02Assertion (A) & Reason (B) MCQ
11 Q→031 Marks
72 Q→042 Marks
37 Q→053 Marks
20 Q→064 Marks
23 Q→07Case study (4 Marks)
31 Q→Sample Questions
Application of Derivatives questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The value of $b$ for which the function $f(x)=x+\cos x+b$ is strictly decreasing over $R$ is
- A$b<1$
- BNo value of $b$ exists
- C$b \leq 1$
- D$b \geq 1$
The interval, in which function $y=x^3+6 x^2+6$ is increasing, is
- A$(-\infty,-4) \cup(0, \infty)$
- B$(-\infty,-4)$
- C$(-4,0)$
- D$(-\infty, 0) \cup(4, \infty)$
The total surface area (S) of the casted half cylinder will be
- A$\pi r h+2 \pi r^2+r h$
- B$\pi r h+\pi r^2+2 r h$
- C$2 \pi r h+\pi r^2+2 r h$
- D$\pi r h+\pi r^2+r h$
The value of $x$ for which $\left(x-x^2\right)$ is maximum, is
- A$3 / 4$
- B$1 / 2$
- C$1 / 3$
- D$1 / 4$
Assertion $( A )$ : The maximum value of the function $f(x)$ $=x^5, x \in[-1,1]$, is attained at its critical point, $x=0$. Reason (R): The maximum of a function can only occur at points where derivative is zero.
- ABoth $(A)$ and $(R)$ are true and $(R)$ is the correct explanation for (A).
- BBoth $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation for (A).
- C(A) is false but $(R)$ is true.
- ✓Both $(A)$ and $(R)$ are false.
Answer: D.
View full solution →Consider the function $f(x)=x^{1 / 3}, x \in R$.
Assertion (A) : $f$ has a point of inflexion at $x=0$.
Reason $( R ): f^{\prime \prime}(0)=0$.
Assertion (A) : $f$ has a point of inflexion at $x=0$.
Reason $( R ): f^{\prime \prime}(0)=0$.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- ✓(A) is true but (R) is false.
- D(A) is false but (R) is true.
Answer: C.
View full solution →Assertion (A) : Let $f: R \rightarrow R$ be a function such that $f(x)=x^3+x^2+3 x+\sin x$. Then, $f$ is an increasing function.
Reason (R) : If $f^{\prime}(x)<0$, then $f(x)$ is a decreasing function.
Reason (R) : If $f^{\prime}(x)<0$, then $f(x)$ is a decreasing function.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- D(A) is false but (R) is true.
Assertion (A) : Both $\sin x$ and $\cos x$ are decreasing functions in $\left(\frac{\pi}{2}, \pi\right)$.
Reason (R): If a differentiable function decreases in $(a, b)$, then its derivative also decreases in $(a, b)$.
Reason (R): If a differentiable function decreases in $(a, b)$, then its derivative also decreases in $(a, b)$.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- D(A) is false but (R) is true.
Assertion (A): If the function $f(x)=\frac{a e^x+b e^{-x}}{c e^x+d e^{-x}}$ is increasing function of $x$, then $b c>a d$.
Reason (R): A function $f(x)$ is increasing if $f^{\prime}(x)>0$ for all $x$.
Reason (R): A function $f(x)$ is increasing if $f^{\prime}(x)>0$ for all $x$.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- ✓(A) is false but (R) is true.
Answer: D.
View full solution →The maximum value of $[x(x-1)+1]^{\frac{1}{3}}, 0 \leq x \leq 1$ is
View full solution →For all real values of x, the minimum value of $\frac{1-x+x^{2}}{1+x+x^{2}}$ is
View full solution →The point on the curve x2 = 2y which is nearest to the point (0, 5) is
Find the interval of the function that is strictly increasing or decreasing: (x + 1)3 (x - 3)3
Find the interval in function 6 - 9x - x2 is increasing or decreasing.
What is the maximum value of the function sin x + cos x?
At what points in the interval [0, 2$\pi$], does the function sin 2x attain its maximum value?
View full solution →Find both the maximum value and minimum value of $3{x^4} - 8{x^3} + 12{x^2} - 48x + 25$ on the interval [0, 3].
View full solution →Find the maximum profit that a company can make, if the profit function is given by
p(x) = 41 – 72x – 18x2
p(x) = 41 – 72x – 18x2
Find the absolute maximum value and the absolute minimum value of the function:
$f(x)=(x-1)^{2}+3, x \in[-3,1]$
View full solution →$f(x)=(x-1)^{2}+3, x \in[-3,1]$
Find the intervals in which the function f given by f(x) = $ x ^ { 3 } + \frac { 1 } { x ^ { 3 } } , x \neq 0$ is x (i) increasing. (ii) decreasing.
View full solution →Find the intervals in which the function f given by $f\left( x \right) = \frac{{4\sin x - 2x - x\cos x}}{{2 + \cos x}}$ is
View full solution →- increasing
- decreasing
Show that the function given by f(x) = $\frac{\log x}{x}$ has maximum at x = e.
View full solution →Find the local maxima and local minima of function,
$f(x)=x \sqrt{1-x}, \quad 0<x<1$
Find also the local maximum and the local minimum value.
View full solution →$f(x)=x \sqrt{1-x}, \quad 0<x<1$
Find also the local maximum and the local minimum value.
Find the local maxima and local minima of function
$g(x)=\frac{1}{x^{2}+2}$
Find also the local maximum and the local minimum value.
View full solution →$g(x)=\frac{1}{x^{2}+2}$
Find also the local maximum and the local minimum value.
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is$(a^{\frac{2}{3}} + b^{\frac{2}{3}})^{\frac{3}{2}}$
View full solution →A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their area is least when the side of square is double the radius of the circle.
A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq.metres for the base and Rs 45 per sq. metre for sides. What is the cost of least expensive tank ?
Find the maximum area of an isosceles $\triangle $ inscribed in the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ with its vertex at one end of the major axis
View full solution →Read the following passage and answer the questions given below.
View full solution →
The temperature of a person during an intestinal illness is given by $f(x)=-0.1 x^2+m x+98.6,0 \leq x \leq 12, \mathrm{~m}$ being a constant, where $\mathrm{f}(\mathrm{x})$ is the temperature in ${ }^{\circ} \mathrm{F}$ at $x$ days.
(i) Is the function differentiable in the interval $(0,12)$ ? Justify your answer.
(ii) If 6 is the critical point of the function, then find the value of the constant $\mathrm{m}$.
(iii) Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
(iii) Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute maximum/absolute minimum values of the function.
An Apache helicopter of the enemy is flying along the curve given by $y=x^2+7$. A soldier, placed at $(3,7)$ want to shoot down the helicopter when it is nearest to him.
View full solution →
(i) If $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ be the position of a helicopter on curve $\mathrm{y}=\mathrm{x}^2+7$, then find distance $\mathrm{D}$ from $\mathrm{P}$ to soldier place at $(3,7)$.
(ii) Find the critical point such that distance is minimum.
(iii) Verify by second derivative test that distance is minimum at $(1,8)$.
OR
Find the minimum distance between soldier and helicopter?
The relation between the height of the plant ( $\mathrm{y}$ in $\mathrm{cm}$ ) with respect to exposure to sunlight is governed by the following equation $\mathrm{y}=4 \mathrm{x}-\frac{1}{2} \mathrm{x}^2$ where $\mathrm{x}$ is the number of days exposed to sunlight.
View full solution →
(i) Find the rate of growth of the plant with respect to sunlight.
(ii) What is the number of days it will take for the plant to grow to the maximum height?
(iii) Verify that height of the plant is maximum after four days by second derivative test and find the maximum height of plant.
OR
What will be the height of the plant after 2 days?
A gardener wants to construct a rectangular bed of garden in a circular patch of land. He takes the maximum perimeter of the rectangular region as possible. (Refer to the images given below for calculations)
View full solution →
(i) Find the perimeter of rectangle in terms of any one side and radius of circle.
(ii) Find critical points to maximize the perimeter of rectangle?
(iii) Check for maximum or minimum value of perimeter at critical point.
OR
If a rectangle of the maximum perimeter which can be inscribed in a circle of radius $10 \mathrm{~cm}$ is square, then the perimeter of region.
Read the following passage and answer the questions given below: elation between the height of the plant $\left(y^{\prime}\right.$ in cm $)$ with respect to its exposure to is governed by the following equation $y=4 x-\frac{1}{2} x^2$, where ' $x$ ' is the number of days exposed to the sunlight, for $x \leq 3$
View full solution →
(i) Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
(ii) Does the rate of growth of the plant increase or decrease in the first three days?
What will be the height of the plant after 2 days?
Generate a Application of Derivatives paper free
Pick question groups from the list above, set marks and difficulty, and export a branded PDF with step-by-step answer keys. First 3 chapters free — no signup.