Question

Suppose \(f''\) is continuous on \(\left( { - \infty ,\infty } \right)\).

(a) If \({f^\prime }(2) = 0,\;{\rm{and}}\;{f^{\prime \prime }}(2) = - 5\), what can you say about \(f\)?

(b) If \({f^\prime }(6) = 0,\;{\rm{and}}\;{f^{\prime \prime }}(6) = 0\), what can you say about \(f\)?

Short answer

(a) The function is continuous, the function \(f\) has a local maximum at the point \(2\).

(b) The function is continuous, the function \(f\) fails to satisfy the Second derivative test and nothing can be obtained by the given information.

Step-by-step solution

Given data

(a) The values \({f^\prime }(2) = 0,{f^{\prime \prime }}(2) = - 5\).

(b) The values are, \({f^\prime }(6) = 0{f^{\prime \prime }}(6) = 0\).

Concept of second derivative test

Second Derivative Test: "Suppose\({f^{\prime \prime }}\)is continuous near\(c\).

(a) If\({f^\prime }(c) = 0\)and\({f^{\prime \prime }}(c) > 0\), then\(f\)has a local minimum at\(c\).

(b) If\({f^\prime }(c) = 0\)and\({f^{\prime \prime }}(c) < 0\), then\(f\)has a local maximum at\(c\).

(c) If\({f^\prime }(c) = 0\)and\({f^{\prime \prime }}(c) = 0\), then the test fails.

(a)Step 3: Determine about the function \(f\) by using the given conditions\({f^\prime }(2) = 0,\;{f^{\prime \prime }}(2) =  - 5\)

Since in the given conditions, the values of the first and second derivative of the function is given, it is easy to justify a function by the Second Derivative Test.

So, compare the given values with the conditions of the Second Derivative Test.

Since \({f^\prime }(2) = 0,\;{f^{\prime \prime }}(2) = - 5 < 0\) and the function is continuous, the function \(f\) has a local maximum at the point \(2\).

This is the description of \(f\) from the given conditions.

(b)Step 4: Determine about the function \(f\) by using the given conditions\({f^\prime }(6) = 0,\;{f^{\prime \prime }}(6) = 0\)

Since in the given conditions, the values of the first and second derivative of the function is given, it is easy to justify a function by the Second Derivative Test. So, compare the given values with the conditions of the Second Derivative Test.

Since \({f^\prime }(6) = 0,\;{f^{\prime \prime }}(6) = 0\) and the function is continuous, the function \(f\) fails to satisfy the Second derivative test and nothing can be obtained by the given information.

This is the description of \(f\) from the given conditions.