Problem
Find the relative extreme points of the function, if they exist. Then sketch a graph of the function. \[ F(x)=\sqrt[3]{x+5} \] Identify all the relative maximum points. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The relative maximum point(s) is/are (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an ordered pair. Use a comma to separate answers as needed.) B. There are no relative maximum points. Identify all the relative minimum points. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The relative minimum point(s) is/are (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an ordered pair. Use a comma to separate answers as needed.) B. There are no relative minimum points.
Solution
Step 1 ：Given the function \(F(x)=\sqrt[3]{x+5}\), we need to find the relative extreme points of the function, if they exist.
Step 2 ：First, we find the derivative of the function. The derivative of \(F(x)=\sqrt[3]{x+5}\) is \(F'(x)=\frac{1}{3(x + 5)^{2/3}}\).
Step 3 ：The critical points of the function are the points where the derivative is either zero or undefined. However, the derivative of this function is never zero or undefined, so the function does not have any critical points.
Step 4 ：Since the function does not have any critical points, it does not have any relative extrema.
Step 5 ：\(\boxed{\text{There are no relative maximum points.}}\)
Step 6 ：\(\boxed{\text{There are no relative minimum points.}}\)
