Physics I / Extrema and optimization
Practice question · Multiple choice

f′(c)=0f'(c) = 0 at an interior point cc. A classmate concludes "ff has a max or min at cc." What is the verdict?

Hints
  1. Find a curve that is flat for one instant in the middle of a climb.
  2. Check f(x)=x3f(x) = x^{3}: its derivative 3x23x^{2} vanishes at 0, yet the function is increasing on both sides.
Show the answer

B. Not necessarily: it can flatten and keep climbing

Why

f′(c)=0f'(c)=0 nominates cc as a candidate; it does not elect it. x3x^{3} pauses flat at the origin mid-ascent, a saddle. Confirmation needs a sign change in f′f' or a second-derivative test.

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