Mathematics

Mathematics

Problem

A continuous function ff on [1,3][1,3] has a local maximum 88 at x=1x=1 and a local minimum −2-2 at x=3x=3, with no other stationary points in the interval. Which of the following describes the range of ff on [1,3][1,3]? a) (−2,8)(-2,8) b) Cannot be determined from the given information c) [−2,8][-2,8] d) [1,3][1,3]
Intuitive explanation

The key insight is that a continuous function on a closed interval must attain all values between its absolute minimum and maximum—this is the Intermediate Value Theorem applied to extrema. Since ff has a local max of 8 at the left endpoint and a local min of −2-2 at the right endpoint with no other critical points in between, the function must descend from 8 to −2-2 somewhere in the interior. Therefore, ff takes on every value in the interval [−2,8][-2, 8].

Intuitive explanation

The key insight is that a continuous function on a closed interval must attain all values between its absolute minimum and maximum—this is the Intermediate Value Theorem applied to extrema. Since ff has a local max of 8 at the left endpoint and a local min of −2-2 at the right endpoint with no other critical points in between, the function must descend from 8 to −2-2 somewhere in the interior. Therefore, ff takes on every value in the interval [−2,8][-2, 8].

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