MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Along a 20-mile stretch of a hiking trail, the ground temperature (in °C) at a distance x miles from the trailhead is modeled by a function T that is continuous on the closed interval [0,20].
Park rangers wish to know information about extreme temperatures on the trail.
(1) Explain why T must attain at least one highest temperature and at least one lowest temperature somewhere along the trail. Be explicit about the conditions that guarantee this result.
(2) State all locations (values of x) where these extreme temperatures are guaranteed to occur. Distinguish between endpoints and interior points, and justify your answer using appropriate definitions.
(3) Suppose, in addition, that T is differentiable on (0,20) and T′(x)=0 only at x=4, x=11, and x=17. Explain how this information refines your answer in part (b).