Locate the critical points of the following function. Then use the Second Derivative Test to determine whether they correspond to local maxima or local minima, or if the test is inconclusive.
[tex]f(x)=\frac{1}{4} x^4-\frac{5}{3} x^3-\frac{25}{2} x^2+125 x[/tex]
A. [tex]x=-5[/tex]
B. There are no critical points that are local minima according to the Second Derivative Test.
Identify all the critical points that are local maxima by the Second Derivative Test. Select the correct choice below and fill in any answer boxes within your choice.
A. [tex]x=[/tex]
B. There are no critical points that are local maxima according to the Second Derivative Test.
Identify all the critical points for which the Second Derivative Test is inconclusive. Select the correct choice below and fill in any answer boxes within your choice.
A. [tex]x=[/tex]
B. There are no critical points for which the Second Derivative Test is inconclusive.
Answer (1)
A. x = − 5 is a local minimum. B. There are no critical points that are local maxima according to the Second Derivative Test. A. x = 5 is a critical point for which the Second Derivative Test is inconclusive.
Examples
Understanding critical points and the second derivative test is crucial in many fields. For example, in economics, businesses use these concepts to find the production level that maximizes profit or minimizes cost. By analyzing the first and second derivatives of cost and revenue functions, companies can identify optimal operating points.
