Mathematics

Mathematics

Problem

Let g(x)=sec⁡xg(x)=\sec x be defined on the closed interval [−π3,π3]\left[ -\dfrac{\pi}{3},\dfrac{\pi}{3} \right]. (a) Verify that gg is continuous on the given closed interval and differentiable on the corresponding open interval. (b) Determine every critical number of gg in (−π3,π3)\left( -\dfrac{\pi}{3},\dfrac{\pi}{3} \right) and use them to locate all local extrema on this interval. (c) Compute g′′(x)g''(x) and analyze its sign to identify the intervals of concavity for gg on (−π3,π3)\left( -\dfrac{\pi}{3},\dfrac{\pi}{3} \right). Clearly state any points of inflection. (d) Combine your conclusions to sketch a detailed graph of y=sec⁡xy=\sec x on the given interval, labeling all key features obtained in parts (b) and (c). Explain how each feature is reflected in your sketch.
Intuitive explanation

The key to this problem is recognizing that sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}, so finding extrema of sec⁡x\sec x depends entirely on where cos⁡x\cos x reaches its minimum and maximum values. Since cosine is even and increasing on [0,π3]\left[0, \frac{\pi}{3}\right], secant will be decreasing there—meaning the extremes occur at the endpoints and center of your interval. Think about which location gives you the largest and smallest reciprocal of cosine.

Intuitive explanation

The key to this problem is recognizing that sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}, so finding extrema of sec⁡x\sec x depends entirely on where cos⁡x\cos x reaches its minimum and maximum values. Since cosine is even and increasing on [0,π3]\left[0, \frac{\pi}{3}\right], secant will be decreasing there—meaning the extremes occur at the endpoints and center of your interval. Think about which location gives you the largest and smallest reciprocal of cosine.

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