Let be defined on the closed interval .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

g(x)=\sec x

be defined on the closed interval

\left[ -\dfrac{\pi}{3},\dfrac{\pi}{3} \right]

. (a) Verify that

g

is continuous on the given closed interval and differentiable on the corresponding open interval. (b) Determine every critical number of

g

\left( -\dfrac{\pi}{3},\dfrac{\pi}{3} \right)

and use them to locate all local extrema on this interval. (c) Compute

g''(x)

and analyze its sign to identify the intervals of concavity for

g

\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 x

on the given interval, labeling all key features obtained in parts (b) and (c). Explain how each feature is reflected in your sketch.