Question

The necessary condition to be maximum or minimum for the function is

• A. $f\text{'}\left(x\right)=0$and it is sufficient
• B. $f\text{'}\text{'}\left(x\right)=0$and it is sufficient
• C. $f\text{'}\left(x\right)=0$but it is not sufficient
• D. $f\text{'}\left(x\right)=0$and $f\text{'}\text{'}\left(x\right)=-ve$

Answer 1

The correct option is C

$f\text{'}\left(x\right)=0$but it is not sufficient

Explanation for the correct option:

Let $f\left(x\right)$ be a function in the range $\left(a,b\right)$ and also having the maximum and minimum values in the range $\left(a,b\right)$.

For calculating maximum or minimum we will first differentiate the given function and check whether the derivative of the function is equal to zero or not.

So we can say that $f\text{'}\left(x\right)=0$ is a necessary condition for calculating maximum and minimum.
After having the value of $f\text{'}\left(x\right)=0$, we need to calculate the value of $f\text{'}\text{'}\left(x\right)$ .

Here, if we get that $f\text{'}\text{'}\left(x\right)<0$ , then the function will have maximum value.
If we get the that $f\text{'}\text{'}\left(x\right)>0$, then the function will have minimum value.

So we can say that the condition $f\text{'}\left(x\right)=0$ is a necessary condition but not sufficient for calculating the maximum or minimum of the function.

Hence, option C is the correct .