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Chapter 3: Problem 22

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

Short Answer

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

Step by step solution

01

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 \(f'(x)=-2(x-5)\).
02

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.
03

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.

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Most popular questions from this chapter

Assume that \(f\) is differentiable for all \(x\). The signs of \(f^{\prime}\) are as follows. \(f^{\prime}(x)>0\) on \((-\infty,-4)\) \(f^{\prime}(x)<0\) on (-4,6) \(f^{\prime}(x)>0\) on \((6, \infty)\) Supply the appropriate inequality for the indicated value of \(c\). $$ g(x)=f(x-10) \quad g^{\prime}(0) \quad 0 $$

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