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?
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
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