Application of DerivativeHard
Question
Let h(x) = f(x) - (f(x))2 + (f(x))3 for every real number x. Then
Options
A.h is increasing whenever f is increasing
B.h is increasing whenever f is decreasing
C.h is decreasing whenever f is decreasing
D.nothing can said in general
Solution
Given h(x) = f(x) - (f(x))2 + (f(x))3
On differentiating w. r. t. x, we get
h′(x) = f′(x) - 2 f(x). f′(x) + 3 f2(x). f′(x)
= f′(x)[1- 2 f(x) + 3 f2(x)]
= 3f′(x)
= 3f′(x)
= 3f′(x)
= 3f′(x)
Note that h′(x) < 0 if f′(x) < 0 and h′(x) > 0 and f′(x) > 0
Therefore, h(x) is increasing function if f (x) is increasing function, and h(x) is decreasing function if f(x) is decreasing function.
Therefore, options (a) and (c) are correct answers.
On differentiating w. r. t. x, we get
h′(x) = f′(x) - 2 f(x). f′(x) + 3 f2(x). f′(x)
= f′(x)[1- 2 f(x) + 3 f2(x)]
= 3f′(x)
= 3f′(x)
= 3f′(x)
= 3f′(x)
Note that h′(x) < 0 if f′(x) < 0 and h′(x) > 0 and f′(x) > 0
Therefore, h(x) is increasing function if f (x) is increasing function, and h(x) is decreasing function if f(x) is decreasing function.
Therefore, options (a) and (c) are correct answers.
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