Question

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

• A. f(x) is an increasing function.
• B. f(x) is a decreasing function.
• C. $f^2\left(x\right)$ is a decreasing function.
• D. |f(x)| is an increasing function.

Answer 1

The correct option is C $f^2\left(x\right)$ is a decreasing function.

Given that,
$f\left(x\right)f\text{'}\left(x\right)<0\Rightarrow \frac{1}{2}\frac{d}{\mathrm{dx}}\left(f^2\left(x\right)\right\}<0\Rightarrow \frac{d}{\mathrm{dx}}\left(f^2\left(x\right)\right\}<0\mathrm{Thus},f^2\left(x\right)\mathrm{is}a\mathrm{decreasing}\mathrm{function}.$