Application of Derivatives: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Find where $f=x^2-4x$ is decreasing.
2. Rate of change of volume of a cube w.r.t. edge at $a=2$.
3. Minimum value of $x^2+4x+7$.
4. Is $f=-x^3$ increasing or decreasing on R?
5. Critical point of $x\log x$.
6. Max value of $3-2\sin x$.
7. Find $\frac{dA}{dt}$ if $A=x^2$ and $\dot{x}=3$ at $x=4$.
8. Local min value of $x+\frac{4}{x}$, $x>0$.
9. Is $e^{-x}$ increasing?
10. Where does $f=\sin x$ have a max on $[0,\pi ]$?

D. Short answer I (2 marks each)

1. A spherical balloon's radius increases at 0.2 cm/s. Find the rate of volume at $r=5$.
2. Find the intervals where $f=x^3-6x^2+9x$ is increasing.
3. Find two positive numbers with sum 20 whose product is largest.
4. Show that $f=\tan x-x$ is increasing on $(0,\frac{\pi }{2})$.
5. Find the local extrema of $f=x^2{e}^{-x}$.
6. A circle's area increases at 6 cm²/s. Find the rate of the radius at $r=3$.
7. Find the absolute max of $f=x^3-3x$ on $[0,2]$.
8. For which $k$ is $f=k{x}^3-9x^2+9x+3$ increasing on R?

Rough work

E. Short answer II (3 marks each)

1. A 10 m ladder slides down a wall; its foot moves away at 1.2 m/s. How fast does the top fall when the foot is 6 m from the wall?
2. Find the intervals of increase/decrease of $f=\sin x+\cos x$, $0\le x\le 2\pi$.
3. Find the points of local max and min of $f=x^4-8x^3+22x^2-24x+5$.
4. A wire of 20 cm is bent into a rectangle. Find the dimensions of maximum area.
5. Water flows into an inverted cone (radius 5 cm, height 10 cm) at 4 cm³/s. Find the rate of rise of level at depth 4 cm.
6. Find the absolute max and min of $f=2\cos x+x$ on $[0,\pi ]$.

F. Long answer (5 marks each)

1. An open box with a square base is to have a volume of 32 m³. Find the dimensions that minimise the material used.
2. Show that of all right triangles with a given hypotenuse, the isosceles one has the greatest area. Find this area for hypotenuse 10 cm.

G. HOTS (higher-order thinking)

1. Can a function have a local max at a point where it is not differentiable? Example.
2. Why does $f(x)=x^3$ have neither max nor min though $f^{\prime }(0)=0$?
3. If $f^{\prime }(x)=(x-1{)}^2(x-2)$, where are the extrema?
4. Explain why a continuous function on an open interval may have no maximum.

20 MCQs

1. Rate of change of $\frac{4}{3}\pi r^3$ w.r.t. $r$ at $r=2$
   (a) $8\pi$ (b) $32\pi$ (c) $16\pi$ (d) $4\pi$
2. $f=x^2-2x$ is decreasing on
   (a) $(-\infty ,1)$ (b) $(1,\infty )$ (c) R (d) $(0,2)$
3. Local min value of $x^2-6x+10$
   (a) 3 (b) 0 (c) 10 (d) 1
4. $x^3$ at $x=0$ has
   (a) a max (b) neither max nor min (c) a min (d) a corner
5. Max value of $\sin x+\cos x$
   (a) 2 (b) $\sqrt{2}$ (c) 1 (d) $\frac{1}{\sqrt{2}}$
6. $f=e^{-x}$ is
   (a) increasing (b) neither (c) constant (d) decreasing
7. Abs max of $x^2$ on $[-1,2]$
   (a) 4 (b) 1 (c) 2 (d) 0
8. Two numbers with sum 10 and max product
   (a) 4, 6 (b) 3, 7 (c) 5, 5 (d) 2, 8
9. $f^{\prime \prime }(c)<0$, $f^{\prime }(c)=0$: $c$ is a local
   (a) min (b) none (c) inflexion (d) max
10. $\frac{d}{dt}(\pi r^2)$ when $\dot{r}=1$, $r=3$
    (a) $3\pi$ (b) $6\pi$ (c) $9\pi$ (d) $2\pi$
11. $\log x$ has
    (a) a max (b) a min (c) no extremum (d) both
12. $x+\frac{1}{x}$, $x>0$: min value
    (a) 2 (b) 1 (c) 0 (d) $\frac{1}{2}$
13. Increasing on R
    (a) $x^2$ (b) $x^3+x$ (c) $\sin x$ (d) $|x|$
14. Critical points of $x^3-3x^2$
    (a) 0, 3 (b) 1, 2 (c) 0, 2 (d) 2 only
15. Square box from sheet of side 12, cut $x$ for max volume
    (a) 3 (b) 6 (c) 4 (d) 2
16. $f=\cos x$ on $(0,\pi )$
    (a) decreasing (b) increasing (c) constant (d) neither
17. Marginal cost if $C=x^2+3x$, at $x=5$
    (a) 13 (b) 40 (c) 8 (d) 25
18. Max of $f=4-(x-2{)}^2$
    (a) 2 (b) $-2$ (c) 0 (d) 4
19. Cylinder of given surface, max volume: $h=$
    (a) $r$ (b) $2r$ (c) $3r$ (d) $\frac{r}{2}$
20. Abs min of $x^3$ on $[-1,1]$
    (a) 0 (b) 1 (c) $-1$ (d) none

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: $f=x^3+2x$ is increasing on R. R: $f^{\prime }(x)=3x^2+2>0$ for all $x$.
2. A: $f=x^3$ has a local max at 0. R: $f^{\prime }(0)=0$.
3. A: The abs max of $x^2$ on $[-2,1]$ is 4. R: abs extrema on a closed interval may occur at end points.
4. A: $\sin x$ is increasing on $(0,\pi )$. R: $\cos x>0$ on $(0,\frac{\pi }{2})$.
5. A: A square has the largest area among rectangles of a given perimeter. R: for $x+y=k$, $xy$ is largest at $x=y$.
6. A: $e^x$ has no maximum on R. R: $e^x>0$ for all $x$.

Key: 1 (a) · 2 (d) · 3 (a) · 4 (d) · 5 (a) · 6 (b)

Answers to the practice set