Question

If the function $f\left(x\right)$ increases in the interval $(a,b)$, then the function $\varphi \left(x\right)=\left[f\right(x){]}^2$

• A. Increases in $(a,b)$
• B. Decreases in $(a,b)$
• C. We cannot say that $\varphi \left(x\right)$ increases or decreases in $(a,b)$
• D. None of the above.

Answer 1

The correct option is C We cannot say that $\varphi \left(x\right)$ increases or decreases in $(a,b)$

$f\left(x\right)$ increases in the interval $(a,b)$

Therefore, $f^{\prime }\left(x\right)>0$

$\varphi \left(x\right)={\left[f\left(x\right)\right]}^2$

$\varphi \left(x\right)=2f\left(x\right)f^{\prime }\left(x\right)$

Now, we dont know about the sign of $f\left(x\right)$

And thus ${\varphi }^{\prime }\left(x\right)$ can either be increasing or decreasing in $(a,b)$