Problem 22 - Essential Calculus

Question

Find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=-(x-5)^{2}\)

Step-by-Step Solution

Answer

Based on these steps, the function has a relative maximum at the point \(x=5\).

Step 1: Find the derivative of the function

First, find the derivative of the function \(f'(x)\). The derivative will yield a function that provides the rate of change of the original function. The derivative of \(f(x)=-(x-5)^{2}\) is \(f'(x)=-2(x-5)\).

Step 2: Find the critical points

The critical points of a function are where the derivative equals zero or is undefined. For \(f'(x)=-2(x-5)\), set \(f'(x)=0\) and solve for \(x\) which results in \(x=5\). Hence, \(5\) is a critical point of the function.

Step 3: Apply the second derivative test

The second derivative test involves finding the second derivative \(f''(x)\), then substituting the critical points found into \(f''(x)\). If \(f''(x)\) is greater than zero at that critical point, it is a local minimum, if it's less than zero, it's a local maximum, and if it's equal to zero, the test is inconclusive. The second derivative of \(f(x)=-(x-5)^{2}\) is \(f''(x) = -2\). Substituting \(x=5\) into \(f''(x)\), we get \(f''(5)=-2\). Since \(f''(5)<0\), \(x=5\) is a relative maximum.