If f is differentiable and the difference quotient overestimates the slope of f at x=a for all h>0, which must be true?

a) f '(a)>0

b) f '(a)<0

c) f ''(a)>0

d) f ''(a)<0

i don't understand the concept of this question. can someone help?

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- Apr 2nd 2008, 08:34 PMgot_janedifferentiation
If f is differentiable and the difference quotient overestimates the slope of f at x=a for all h>0, which must be true?

a) f '(a)>0

b) f '(a)<0

c) f ''(a)>0

d) f ''(a)<0

i don't understand the concept of this question. can someone help? - Apr 2nd 2008, 10:33 PMCaptainBlack
You are told that:

$\displaystyle \frac{f(a+h)-f(a)}{h}>f'(a), \ \forall h>0$.

But the answer is none of the the offered options, for consider $\displaystyle f(x)=x^2$, and $\displaystyle a=0$ , the difference quotient is:

$\displaystyle \frac{h^2}{h}=h>0, \ \forall h>0$

but $\displaystyle f'(a)=0$, and $\displaystyle f''(a)=0$.

RonL - Apr 2nd 2008, 10:41 PMh2osprey
- Apr 3rd 2008, 03:00 AMCaptainBlack

Brain fart on my part, try f(x)=x^3, f'(x)=3x^2, f''(x)=6x, so the difference quotient at x=0 is h^3>0 for h>0, but the first two derivatives are zero.

(the problem arrose because of a last minute replacement of a much more interesting counter example, the point was that you can construct a function satisfying the required condition but which satisfy none of the given candidates for the answer, as an alternative consider f(x)=x^4)

In fact the conditions on f'' cannot apply even if assorted ">"s are changed to ">="'s because there is no condition requiring f'' to exist.

Now there may be a formulation of a question like this that does have an answer like c, but this is not that question.

RonL - Apr 3rd 2008, 02:06 PMgot_jane
thank you so much! the both of you!(Handshake)