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There exists a differentiable function f (-1, 1)→ R that does not have any local extrema but whose derivative vanishes at one point in (-1, 1). Select one: Ⓒa. True, here is an example: f(x) = x³. O b. True, because f is continuous on (-1, 1) which is an interval and thus it has a maximum and a minimum on that interval, and its derivative vanishes at these points by the IET. O c. False, because by the IET if f'(c) = 0 with c E (-1, 1) then c is at least a local extremum of f. O d. True, here is an example: f(x) = sin(x).

There exists a differentiable function f (-1, 1)→ R that does not have any local extrema but whose derivative vanishes at one point in (-1, 1). Select one: Ⓒa. True, here is an example: f(x) = x³. O b. True, because f is continuous on (-1, 1) which is an interval and thus it has a maximum and a minimum on that interval, and its derivative vanishes at these points by the IET. O c. False, because by the IET if f'(c) = 0 with c E (-1, 1) then c is at least a local extremum of f. O d. True, here is an example: f(x) = sin(x).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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There exists a differentiable function f (-1, 1)→ R that does not have any local extrema but whose derivative vanishes at one point in (-1, 1). Select one: Ⓒa. True, here is an example: f(x) = x³. O b. True, because f is continuous on (-1, 1) which is an interval and thus it has a maximum and a minimum on that interval, and its derivative vanishes at these points by the IET. O c. False, because by the IET if f'(c) = 0 with c E (-1, 1) then c is at least a local extremum of f. O d. True, here is an example: f(x) = sin(x).
Transcribed Image Text:There exists a differentiable function f (-1, 1)→ R that does not have any local extrema but whose derivative vanishes at one point in (-1, 1). Select one: Ⓒa. True, here is an example: f(x) = x³. O b. True, because f is continuous on (-1, 1) which is an interval and thus it has a maximum and a minimum on that interval, and its derivative vanishes at these points by the IET. O c. False, because by the IET if f'(c) = 0 with c E (-1, 1) then c is at least a local extremum of f. O d. True, here is an example: f(x) = sin(x).
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