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Chapter 4: Problem 68

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$h(t)=2+\cos 2 t, \text { for }-\pi \leq t \leq \pi$$

Short Answer

Expert verified
Answer: The function is concave up on the intervals \((-\pi, -\frac{3\pi}{4})\), \((-\frac{\pi}{4}, \frac{\pi}{4})\), and \((\frac{3\pi}{4}, \pi)\). The function is concave down on the intervals \((-\frac{3\pi}{4}, -\frac{\pi}{4})\) and \((\frac{\pi}{4}, \frac{3\pi}{4})\). The inflection points occur at \(t = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}\).

Step by step solution

01

Find the first and second derivatives of the given function

To find the first derivative, apply the chain rule: $$h'(t) = \frac{d}{dt} (2+\cos 2t) = 0 -2\sin(2t)\cdot \frac{d}{dt}(2t) = -4\sin(2t)$$ To find the second derivative, differentiate the first derivative with respect to t: $$h''(t) = \frac{d}{dt} (-4\sin(2t)) = -8\cos(2t)$$
02

Find the critical points

Set the second derivative equal to zero and solve for t: $$h''(t) = -8\cos(2t) = 0$$ $$\cos(2t) = 0$$ For the given domain \(-\pi \leq t \leq \pi\), we have the critical points at \(t = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}\).
03

Determine the concavity

Use the second derivative test to determine the concavity on each interval: For \(-\pi \leq t \leq -\frac{3\pi}{4}, h''(t) > 0\), so the function is concave up. For \(-\frac{3\pi}{4} \leq t \leq -\frac{\pi}{4}, h''(t) < 0\), so the function is concave down. For \(-\frac{\pi}{4} \leq t \leq \frac{\pi}{4}, h''(t) > 0\), so the function is concave up. For \(\frac{\pi}{4} \leq t \leq \frac{3\pi}{4}, h''(t) < 0\), so the function is concave down. For \(\frac{3\pi}{4} \leq t \leq \pi, h''(t) > 0\), so the function is concave up.
04

Write the intervals of concavity and inflection points

The function is concave up on the intervals \((-\pi, -\frac{3\pi}{4})\), \((-\frac{\pi}{4}, \frac{\pi}{4})\), and \((\frac{3\pi}{4}, \pi)\). The function is concave down on the intervals \((-\frac{3\pi}{4}, -\frac{\pi}{4})\) and \((\frac{\pi}{4}, \frac{3\pi}{4})\). The inflection points occur at \(t = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}\).

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