Application of Derivatives Test

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Application of Derivatives Test - 18

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Weekly Quiz Competition

Question 1
1 / -0

If $$f(x)=kx^3 -9x^2 +9x+3$$ is increasing for every real number $$x$$, then

A
$$k > 3$$

B
$$k \ge 3$$

C
$$k < 3$$

D
$$k \le 3$$

Solution
Question 2
1 / -0

The tangent to the curve $$y=e^{kx}$$ at a point (0, 1) meets the x-axis at (q, 0) where $$a \epsilon \left [ -2,-1 \right ]$$ then $$k \epsilon $$

A
[-1/2,0]

B
[-1,-1/2]

C
[0,1]

D
[1/2, 1]

Solution
Question 3
1 / -0

A curve $$y=me^{mx}$$ where $$m > 0$$ intersects y-axis at a point $$P$$.
What is the equation of tangent to the curve at $$P$$ ?

A
$$y=mx+m$$

B
$$y=-mx+2m$$

C
$$y=m^2x+2m$$

D
$$y=m^2x+m$$

Solution

$$y = m {e}^{mx}, \; m > 0$$
Substituting $$x = 0$$, we have
$$y = m {e}^{m \cdot 0} = m$$
Therefore,
Slope $$= \cfrac{dy}{dx} = {m}^{2} {e}^{mx} = {m}^{2} {e}^{m \cdot 0} = {m}^{2}$$
Therefore, equation of tangent at $$P$$ is given by-
$$y - m = \cfrac{dy}{dx} \left( x - 0 \right)$$
$$\Rightarrow y - m = {m}^{2} x$$
$$\Rightarrow y = {m}^{2}x + m$$
Question 4
1 / -0

The function $$f(x)=x^3-27x+8$$ is increasing when

A
$$|x| < 3$$

B
$$|x| > 3$$

C
$$-3 < x < 3$$

D
$$none\ of\ these$$

Solution

$$f'(x)=3x^2 -27=3(x^2 -9)=3(x+3)(x-3)$$

$$\therefore \ f'(x)$$ is increasing when $$x < -3$$ or $$x > 3$$ i.e when $$|x| > 3$$.

There are two factors in $$f'(x)$$. so we start with $$+ve$$ sign

$$\therefore \ f(x)$$ is increasing when $$x < -3$$ or $$x > 3$$ i.e when $$|x| > 3$$.
Question 5
1 / -0

If the slope of the tangent to the curve $$xy+ ax+ by=0$$ at the point $$(1, 1) $$ on it is $$2$$, then values of $$a$$ and $$b$$ are

A
$$1, 2$$

B
$$1, -2$$

C
$$-1, 2$$

D
$$-1, -2$$

Solution

$$a+b +1=0$$
$${xy}'+y+a+{b y}'=0$$
$${y}'=\dfrac{-(y+a)}{(x+b )}$$
$$-\dfrac{(a+1)}{(1+b )}=2$$
$$2+2b +a+1=0$$
$$2+2 b -b =0$$
$$b =-2$$
$$a=1$$
Question 6
1 / -0

If the slope of the tangent to the curve $$y = x^{3}$$ at a point on it is equal to the ordinate of the point then the point is

A
$$(27, 3)$$

B
$$(3, 27)$$

C
$$(3, 3)$$

D
$$(1, 1)$$

Solution

Differntiating the given equation
$$\dfrac{dy}{dx}=3x^{2}$$
$$3a^{2}=a^{3}$$
$$a=3$$
$$(3,3^{3})$$
$$(3,27)$$
Question 7
1 / -0

A stationary point of $$\mathrm{f}(\mathrm{x})=\sqrt{16 -\mathrm{x}^{2}}$$ is

A
(4, 0)

B
(-4, 0)

C
(0, 4)

D
(-4, 4)

Solution

$$f(x)=\sqrt {16-x^2}$$
$$y=\sqrt {16-x^2}$$
On $$a$$ starting point $$\dfrac {dy}{dx}=0$$
then $$\dfrac {dy}{dx}=\dfrac {1}{2} \dfrac {-2x}{\sqrt {16-x^2}}=\dfrac {x}{\sqrt {16-x^2}}=0$$
At $$x=0$$
Now $$y=\sqrt {16-0}$$
$$=\sqrt {16}=4$$
$$(0,4)\ $$ be starting point.
Question 8
1 / -0

The critical point of $$\mathrm{f}(\mathrm{x})=|2\mathrm{x}+7|$$ at $$\mathrm{x}=$$

A
$$0$$

B
$$7$$

C
$$\dfrac{-7}{2}$$

D
$$-7$$

Solution

$$f(x)=\mid 2x+7\mid$$
So critical point is at $$x=\frac {-7}{2}$$
Question 9
1 / -0

The number of ciritical points of $$\displaystyle \mathrm{f}(\mathrm{x})=\frac{|x-1|}{x^{2}}$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$0$$

Solution

$$\displaystyle \mathrm{f}(\mathrm{x})=\dfrac{|x-1|}{x^{2}}$$
$$\displaystyle f'(x)=\dfrac{x^{2}(x-1)-2x|x-1|^{2}}{x^{4}|x-1|}$$
For maxima or minima,
$$f'(x)=0$$
$$\Rightarrow (x-1)(-x^2+2x)=0$$
$$\Rightarrow x=0,1,2$$
Hence, number of critical points of $$f(x)$$ are 3.
Question 10
1 / -0

A stationary value of $$\mathrm{f}(\mathrm{x})=\mathrm{x}(\ln \mathrm{x})^{2}$$ is

A
$$2\mathrm{e}^{-2}$$

B
$$4\mathrm{e}^{-2}$$

C
$$2\mathrm{e}^{2}$$

D
$$4\mathrm{e}^{2}$$

Solution

$$f(x)=x(\ln x)^2$$
$$\Rightarrow f'(x)=(\ln x)^2+2x \dfrac {\ln x}{x}$$
$$\Rightarrow f'(x)=\ln x(\ln(x)+2)$$
For stationary points ,
$$f'(x)=0$$
$$\Rightarrow x=1$$ or $$e^{-2}$$
$$f(1)=0$$
and $$f(e^{-2})=4e^{-2}$$

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Re-Attempt Weekly Quiz Competition

Application of Derivatives Test - 18

Result

Application of Derivatives Test - 18

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SHARING IS CARING

Question 1

$$k > 3$$

$$k \ge 3$$

$$k < 3$$

$$k \le 3$$

Solution

Question 2

[-1/2,0]

[-1,-1/2]

[0,1]

[1/2, 1]

Solution

Question 3

$$y=mx+m$$

$$y=-mx+2m$$

$$y=m^2x+2m$$

$$y=m^2x+m$$

Solution

Question 4

$$|x| < 3$$

$$|x| > 3$$

$$-3 < x < 3$$

$$none\ of\ these$$

Solution

Question 5

$$1, 2$$

$$1, -2$$

$$-1, 2$$

$$-1, -2$$

Solution

Question 6

$$(27, 3)$$

$$(3, 27)$$

$$(3, 3)$$

$$(1, 1)$$

Solution

Question 7

(4, 0)

(-4, 0)

(0, 4)

(-4, 4)

Solution

Question 8

$$0$$

$$7$$

$$\dfrac{-7}{2}$$

$$-7$$

Solution

Question 9

$$1$$

$$2$$

$$3$$

$$0$$

Solution

Question 10

$$2\mathrm{e}^{-2}$$

$$4\mathrm{e}^{-2}$$

$$2\mathrm{e}^{2}$$

$$4\mathrm{e}^{2}$$

Solution

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