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Q18E

Expert-verifiedFound in: Page 209

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f\). (Use the graphs and transformations of Sections 1.2.)**

**18. \(f(t) = \cos t,\;\frac{{ - 3\pi }}{2} \le t \le \frac{{3\pi }}{2}\)**

The absolute and local maximum occurs at \(t = 0\). The absolute and local minimum occurs at \(t = \pm \pi \).

The given function is \(f(t) = \cos t,\frac{{ - 3\pi }}{2} \le t \le \frac{{3\pi }}{2}\).

**Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.**

First draw the graph of the function \(f(t) = \cos t\) as shown below in Figure:

Since the function is restricted to \(\frac{{ - 3\pi }}{2} \le t \le \frac{{3\pi }}{2}\), draw the graph of the function in the closed interval of \(\left( {\frac{{ - 3\pi }}{2},\frac{{3\pi }}{2}} \right)\) as shown below in Figure 2 :

Generally the cosine curve attains the maximum at even multiples of \(\pi \) and attains the minimum at odd multiples of \(\pi \) as shown in Figure 1. According to the given interval, from Figure 2, it is observed that the cosine curve attains an absolute maximum at \(t = 0\) and attains an absolute minimum at \(t = \pm \pi \). Therefore, the absolute maximum occurs at \(t = 0\). Since any absolute maximum can be a local maximum, the local maximum is \(f(0) = 1\).

Similarly, the absolute minimum occurs at \(t = \pm \pi \). Since any absolute minimum can be a local minimum, the local minimum is \(f( \pm \pi ) = - 1\).

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