# A function f[x] and a point x=a. If f'[a]>0, is it possible f[a]<=f[x]

You've got a function f[x] and a point x=a.
If f'[a]>0, is it possible that f[a]<=f[x] a for all other x's?
Why?

Having a problems visualize this, any help would be great

## Answers and Replies

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If f'[a] is positive, the function is increasing at a. For f[a]<=f[x] to be true, "a" needs to be a maxima.

Poncho said:
If f'[a] is positive, the function is increasing at a. For f[a]<=f[x] to be true, "a" needs to be a maxima.
Do you mean minima?

...and by the way, if it was a minima, f'(a) would be 0. So since f'(a)>0, what does that tell you about x<a?

I good example would be $$f(x) = \sin x$$ the derivative of which is $$f'(x) = \cos x$$. If $$a = 0$$ then $$f'(0) = \cos 0 = 1$$ indicating the function is rising. However, since sine is a repeating function, it will go to a maximum of 1, and fall down again. So f(a) <= f(x) for all x > a is not true if f(x) repeats on the segment a < x < inf.