Q. 2 TB - Calculus

Question

Second-Derivative Test: Review the second-derivative test for functions of a single variable. Explain how the test works, what conditions a function must satisfy to make the test useful, and when, if ever, the test might fail.

Step-by-Step Solution

Answer

The second derivative test works on the concept of differentiation. It fails when $$f'(k)=0$$ and $$f''(k)=0$$ and the point at $$x=k$$ is the point of inflection.

Step 1. Given Information

Second-derivative test for functions of a single variable. 

Step 2. Explanation

The second derivative test is used to systematically determine the absolute maximum and absolute minimum of a single variable function.

It works on the concept of differentiation. 

To perform the second derivative test of a single variable function, $$f(x)$$ and differentiate it twice at $$x=k$$.

Use the first and second derivatives of the single variable function to determine the local maximum and local minimum.

The conditions a function must satisfy to make the test useful are as follows,

• If $$f'(k)=0$$ and $$f"(k)<0$$, then the point of local maxima is at $$x=k$$ and the local maximum is $$f(k)$$.
• If $$f'(k)=0$$ and $$f"(k)>0$$, then the point of local minima is at $$x=k$$ and the local minimum is $$f(k)$$.

The second derivative test will fail at a condition where $$f'(k)=0$$ and $$f"(k)=0$$ and then $$x=k$$ will be the point of inflection.