# Sketch  the graph of a continuous function f on [0; 5] such that f has no local minimum or maximum, but 1 and 4 are critical points. No need to write a formula for f.

## Expert Answers

A function has a critical point at `x=a` if the derivative `f'(a)=0` or does not exist.  Since the function is continuous, then the points are critical points and not minimums or maximums, then the function has critical points when the curve is horizontal at `x=a` but the second derivative `f''(x)=0` ...

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A function has a critical point at `x=a` if the derivative `f'(a)=0` or does not exist.  Since the function is continuous, then the points are critical points and not minimums or maximums, then the function has critical points when the curve is horizontal at `x=a` but the second derivative `f''(x)=0` also, which means that the function does not change concavity at either `x=1` or `x=4` .

A function of this form would have first derivative like `f'(x)=(x-1)^2(x-4)^2P(x)` where P(x) is any polynomial or other continuous function.

An example of such a function is given in the graph below:

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