Application of Derivatives Test

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Application of Derivatives Test - 9

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Question 1
1 / -0

The function f (x) = x³ - 3x has a

A
local maxima at x = 1

B
local minima at x = 1

C
point of inflexion at 0

D
none of these

Solution

Given, f(x) = x³ - 3x

f'(x) = 3x² - 3

For point of inflexion we have f'(x) = 0

f ′ (x) = 0 ⇒ 3x²− 3 = 0 = 3(x − 1)(x + 1) ⇒ x = ±1

Hence, f(x) has a point of inflexion at x = 0.

When , x is slightly less than 1, f'(x) = (+)(-)(+) i.e, negative

When x is slightly greater than 1, f'(x)= (+)(+)(+) i.e, positive

Hence, f'(x) changes its sign from negative to positive as x increases through 1 and hence x = 1 is a point of local minimum.
Question 2
1 / -0

Let f (x) be differentiable in (0, 4) and f (2) = f (3) and S = {c : 2 < c < 3, f’ (c) = 0 } then

A
S = { }

B
S has atleast one point

C
S has exactly one point

D
none of these

Solution

Since, given f(x) is differentiable in (2,3) and f(2) = f(3) we have conditions of Rolle's theorem are satisfied by f(x) in [2,3].

Hence, there exist atleast one real c in (2,3) s.t.f'(c) = 0.

Therefore, the set S contains atleast one element.
Question 3
1 / -0

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 (x) is continuous for −2 ⩽ x ⩽ 2

C
f ( – 2) = f (2)

D
none of these
Solution
Rolle's theorm states that if
1. f(x) is continuous in [a,b]
2. f(x) is differentiable in(a,b)
3. f(a)=f(b)
Then,there exists at-least one value c in (a,b) such that f'(c)=0

Since f(x)=|x| is not differentiable at x=0 ∈ [-4,4] .Therefore,f(x) is not differentiable in[-4,4] .Thus,one of the conditions of Rolle's theorm is not satisfied by f(x) and hence Rolle's theorm is not applicable to the given function.