Application of Derivatives Test

Result

Application of Derivatives Test - 4

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Question 1
1 / -0

Let f (x) = x⁴ – 4x, then

A
f is increasing in [1,∞)

B
f is decreasing in [1,∞)

C
f is increasing in [−∞,1)

D
none of these

Solution

f (x) = x⁴ – 4x

f'(x) = 4(x³) - 4 = 4(x ³- 1) = 4 {(x²) + x + 1)} (x-1)
= f ‘(x) > 0,if,(x - 1) > 0,
and f ‘(x) <0, if (x -1 ) < 0.
So,f is decreasing on ( - ∞,1] and f is increasing on [1,∞)
Question 2
1 / -0

Let g (x) be continuous at ‘a’ and g(a) ≠ 0. Let f be a function such that f ‘(x) = g(x)(x - a)² , then

A
f is increasing at a if g (a) > 0

B
f is increasing at a if g (a) < 0

C
f is decreasing at a if g (a) >0

D
none of these

Solution

Since g is continuous at a , therefore , if g(a) > 0, then there is a neighbourhood of a, say (a - e, a + e) in which g ( x ) is positive. This means that f ‘(x) >0 in this neighbourhood of a and Hence, f (x) is increasing at a.