Applications Of Derivatives Test

Result

Applications Of Derivatives Test - 2

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Question 1
1 / -0

Let f(x) = xⁿ⁺¹ + axⁿ, where ‘a’ is a positive real number. Then x=0 is a point of

A
local minimum if n is an odd integer

B
local maximum for any integer n

C
local minimum for any integer n

D
local minimum if n is an even integer

Solution

f′(x) = x^n-1 [(n + 1)x + a.n], fⁿ(x) = Ax + B
(where A and B are constants)

So if n is even then f⁽ⁿ⁾ (0) > 0

So local minimum occurs at x=0.

Hence '0' is a point of minimum when n is even.
Question 2
1 / -0

Let f(x) be a function such that f′(x) = log 1/3 (log₃ (sin x + a)). If f(x) is decreasing for all real values of x, then

A
a ∈ (4, ∞)

B
a ∈ (2, 3)

C
a ∈ (1, 4)

D
a ∈ (2, ∞)

Solution

If f(x) is decreasing then f'(x)<0

⇒ log_1⁄3 (og₃ (sin x+a)) < 0

⇒ log₃ (sin x + a) >1

sin x +a>3 a>3-sin x a>4

⇒ a ∈ (4,∞)