Bartlett Ruiz
07/20/2023 · High School
Use the derivative \( f^{\prime} \) to determine the local minima and maxima of \( f \) and the intervals of increase and decrease. Sketch a possible graph of \( f \) ( \( f \) is not unique). \( f^{\prime}(x)=21 \sin 3 x \) on \( \left[-\frac{4 \pi}{3}, \frac{4 \pi}{3}\right] \)
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The critical points are \( x = -\frac{4\pi}{3}, -\pi, -\frac{2\pi}{3}, -\frac{\pi}{3}, 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi \). The function increases in the intervals \( \left[-\frac{2\pi}{3}, -\frac{\pi}{3}\right), \left[-\frac{\pi}{3}, 0\right), \left[\frac{2\pi}{3}, \pi\right), \left[\pi, \frac{4\pi}{3}\right] \) and decreases in \( \left[-\frac{4\pi}{3}, -\pi\right), \left[-\pi, -\frac{2\pi}{3}\right), \left[0, \frac{\pi}{3}\right), \left[\frac{\pi}{3}, \frac{2\pi}{3}\right) \). Local maxima occur at \( x = -\frac{2\pi}{3}, \frac{2\pi}{3} \) and local minima at \( x = -\pi, 0, \pi \).
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