If f '(x) 0 for all x in (-∞,∞), then we know that:

a. f(x) can have no relative extrema.
b. f(x) can have no inflection points.
c. For some x-value, f(x) 0.
d. None of the above are true.

Question

If f '(x)> 0 for all x in (-∞,∞), then we know that:

a. f(x) can have no relative extrema.
b. f(x) can have no inflection points.
c. For some x-value, f(x) > 0.
d. None of the above are true.

mathmagician · Answer

b

anonymous · Answer

thanks!

anonymous · Answer

Disagree with mathmagician. Here's my counterexample:
f'' = 2x
f' = x^2 +1
f = (1/3)x^3 + x

f' Is always positive, but there's an inflection point at x=0.

anonymous · Answer

However, consider what happens to the derivative of a function at its relative extrema.

anonymous · Answer

a

anonymous · Answer

since we'll never have f'(x)=0, there can be no critical points such that f'(x)=0, so there will be no extremes.

anonymous · Answer

And as a counter example for c, consider:
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