Applications of Derivatives Test

Result

Applications of Derivatives Test - 6

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Question 1
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The function f (x) = m x + c where m, c are constants, is a strict decreasing function for all x ∈R if

A
m >0

B
m <0

C
m ⩾0

D
m = 0

Solution

f (x) = mx + c is strict decreasing on R
if f ‘(x) <0 i.e. if m <0 .
Question 2
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The function f (x) = x² −2x is strict decreasing in the interval

A
(−∞, 1]

B
R

C
[1,∞)

D
none of these

Solution

f ‘(x) = 2x –2 = 2 (x - 1) <0 if x <1 i.e. x x ∈(−∞,1) x ∈(−∞,1). Hence f is strict decreasing in left decreasing in (−∞,1)
Question 3
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The points on the curve 4 y = |x² −4| at which tangents are parallel to x –axis, are

A
(4, 3) and (–4, –3)

B
(0, 1) only

C
(2, 0) and (–2, 0)

D
none of these

Solution

Only when x = 0 and when x = 0 , y = 1. So, only at (0,1) tangent is parallel to x- axis.
Question 4
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The maximum value of

A
0

B
–e

C
1/e

D
none of these

Solution

i.e. x = e. Note that , f ‘(x) changes sign from positive to negative as we move from left to right through e. So, f (e) is maximum i.e. maximum value of f (x) is f (e)
Question 5
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Every continuous function is

A
not necessory differentiable

B
not decreasing

C
decreasing

D
differentiable.

Solution

Obviously, every differentiable function is continuous but every continuous function isn 't differentiable.
Question 6
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The function f(x) = tan⁻¹ x is

A
neither increasing nor decreasing

B
differentiable nowhere.

C
strict decreasing

D
strict increasing

Solution

Therefore , f is strictly increasing on R.
Question 7
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For the curve x = t² −1,y = t² −t tangent is parallel to X –axis where

A
t =

B
t =1/2

C
t = 0

D
t = -

Solution
Question 8
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The slope of the normal to the curvex = a (cos θ+ θsin θ),y = a (sin θ–θcos θ) at any point ‘θ’is

A
–cot θ.

B
tan θ

C
–tan θ

D
cot θ

Solution
Question 9
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A

B
has a maximum value 2

C
has a minimum value 0

D

Solution
Question 10
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Rolle ’s Theorem is not applicable to the function f(x) = | x | for −2 ⩽x ⩽2 because

A
f (x) is not derivable for x = 0

B
f (–2) = f (2)

C
f (x) is continuous for −2 ⩽x ⩽2 because

D
none of these

Solution

which does not exist at x = 0 ∈(-2 , 2). So , Rolle ’s theorem is not applicable.
Question 11
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Let f (x) = x³ −6x² +9x+8, then f (x) is decreasing in

A
(−∞,1) ∪(3,∞)

B
[1, 3]

C
[3,∞]

D
(−∞,1)

Solution
Question 12
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The equation of the normal to the curve y = sinx at (0, 0) is

A
y = 0

B
x + y = 0

C
x = 0

D
x –y = 0

Solution

therefore , slope of tangent at (0 , 0) = cos 0 = 1 and hence slope of normal at (0 , 0) is - 1 .
Question 13
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The curve y = x^1/5 has at (0, 0)

A
oblique tangent

B
a vertical tangent

C
no tangent.

D
a horizontal tangent

Solution

so,at (0 ,0) , the curve y = x1/5 has a vertical tangent.
Question 14
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A
a local maxima at x =2 and a local minima at x = –2

B
local minima at x = 2 and a local maxima at x = –2

C
absolute minima at x = 2 and absolute maxima at x = –2.

D
absolute maxima at x = 2 and absolute minima at x = –2

Solution

So, f has a local minima at 2 and a local maxima at - 2 .
Question 15
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A
tan x

B
sin x

C
x

D
none of these

Solution