Question

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

1. ${{f}^{'}}\left( x \right)=0$ and it is sufficient
2. ${{f}^{''}}\left( x \right)=0$ and it is sufficient
3. ${{f}^{'}}\left( x \right)=0$ but it is not sufficient
4. ${{f}^{'}}\left( x \right)=0$ and ${{f}^{''}}\left( x \right)=-ve$

Answer 1

In this problem we need to find the necessary condition for a function to be maximum or minimum. We know that maximum of a function is the maximum value of the function defined in its own range as well as minimum of the function is the minimum value of the function in its range. First we will see how the maximum and minimum of a function are calculated and check the given option for the correct answer.

Complete step by step solution:
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 ${{f}^{'}}\left( x \right)=0$ is a must and should condition for calculating maximum and minimum.
After having the value of ${{f}^{'}}\left( x \right)=0$, we need to calculate the value of ${{f}^{''}}\left( x \right)$ .
Here if we get the value ${{f}^{''}}\left( x \right)$ as negative that means ${{f}^{''}}\left( x \right)<0$ , then the function will have maximum value.
If we get the value ${{f}^{''}}\left( x \right)$ as positive that means ${{f}^{''}}\left( x \right)>0$, then the function will have minimum value.
So we can say that the condition ${{f}^{'}}\left( x \right)=0$ is a necessary condition but not sufficient for calculating the maximum or minimum of the function.
Hence option 3 is the correct answer.

Note: There is a lot of terminology and concepts that are related to maximum or minimum of the function. We can call the maximum or minimum points of the function as extreme points or extremities. Here we have just said that the first order derivative of the function equals zero. But it is not that much correct. Since there is no possibility to have a derivative of a function as zero. We need to calculate the solution for the first order derivative of the function.