First slide

Monotonicity

Question

Let $f : R \to R$ be a differentiable function for all values of x and has the property that f (x) and f '(x) have opposite signs for all values of x. Then,

Easy

A

f (x) is an increasing function
B

f (x) is a decreasing function
C

f²(x) is a decreasing function
D

$| f (x) |$ is an increasing function

Solution

$f (x) f^{'} (x) < 0 \forall x \in R or \frac{1}{2} \frac{d}{dx} (f^{2} (x)) < 0 or \frac{d}{dx} (f^{2} (x)) < 0$
Thus,f²(x) is a decreasing function.

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