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Home > CALC > Chapter 6 > Lesson 6.4.3 > Problem 6-163


If is a differentiable function such that for all real numbers and if , determine if and are relative minimums or maximums. Justify your answer.

Determine if is an extrema candidate.
That is, does or does

(some negative value)  
Yes, is an extrema candidate! It could be a relative max, a relative min, or neither.

Determine if is a relative max, relative min or neither. Use the 1st-derivative test:
To do this, determine if changes sign at .
Evaluate a little to the left and a little to the right.
To the left: (some negative value) (negative)(negative) positive.
To the right: (some negative value) (positive)(negative) negative.
Conclusion: Since and changes from positive to negative at , therefore is a relative maxima.

Now test .