Question

If $f\text{'}\left(x\right)$ is zero in the interval $(a,b)$ then in this interval it is

• A. Increasing function
• B. Decreasing function
• C. Only for $a>0$ and $b>0$, $f\left(x\right)$ is increasing function
• D. None of the above

Answer 1

The correct option is D

None of the above

Explanation for correct option:

It is given that $f\text{'}\left(x\right)$ is zero in the interval $(a,b)$.

$$\begin{gathered} \therefore \forall x\in \left(a,b\right);f\text{'}\left(x\right)=0 \\ \Rightarrow \frac{d}{dx}f\left(x\right)=0 \\ \Rightarrow \int \frac{d}{dx}f\left(x\right)dx=\int 0dx \\ \Rightarrow f\left(x\right)=0+C \\ \Rightarrow f\left(x\right)=C \end{gathered}$$

$\Rightarrow f\left(x\right)$ is a constant function.

We know that if a function is monotonic, then its value must change according to its monotonicity.

But we obtained $f\left(x\right)=C,\forall x\in \left(a,b\right)$. i.e., $f\left(x\right)$ is a constant function.

So, the options A, B, and C are incorrect.

Hence, the correct option is D.