Question

Suppose f"(x)>0 for all x. Which is larger: f(3) or f(4)? f'(3) or f'(4)? f(3.1) or f(3)+f'(3)(0.1)

I believe it's an application of the Mean Value Theorem, but I could be wrong as I am unsure how to do this.

Answer 1

Since \(f''(x)>0\), you know that the first derivative is a strictly increasing function, so you can answer the second question right away.

Answer 2

just as f' is the change in f(x) with x
f'' is the change in f' with x
if f'' is positive that means the change in f' is increasing (i.e. the slope of the function is increasing)

Answer 3

Given the function then,
If for all \[x \ge 0\] in some interval I then is concave up on I.
--------------------
If for all \[x \le 0\] in some interval I then is concave down on I.

There fore f(4) > f(3), same goes for first derivatives

Answer 4

f(x) * is concave up on I ***
f(x) * is concave down on I ***

Answer 5

however, notice for a parabola the slope is very negative and is always increasing as we move to the right along the x axis.

Answer 6

For the first pairing. You could have a function that looks like this

which is concave up \((f''(x)>0)\) and the derivative is increasing (approaching 0 from a negative value) with \(f(3)>f(4)\).

Answer 7

However you could have

so I don't think a conclusion can be reached...