What can we say about the function if its derivative is Strictly increasing.

Question

Let we have a function $f:R\rightarrow R$ such that $f'(x)$ is Strictly increasing. Let $a$ and $b$ denotes the Minimum and maximum on the intervals $[2,3]$. Then, is $b=f(3)$ true? This can only happen when the function is Strictly increasing in the given Interval. But, can we say the Strictly increasing nature of derivative can be extended to say that function is also Strictly increasing?

I am not able to understand the relationship between the function and the monotonicity of its derivative. One thing is sure that it's second derivative is greater that zero. Any help would be beneficial for me. Thanks

Answer 1

Hint: What if the derivative is negative on the entire interval but increasing? For instance, say $f'(x)=x-5$.

Bonus: Does that mean that $f(a)$ and $f(b)$ are always going to be local extrema?

Answer 2

I don't think it's true. Take for example the function $f(x)=e^{-x}$ then $f''(x)=e^{-x} >0,$ thus $f'(x)$ is strictly increasing. Whereas $f(x)$ is strictly decreasing. And $b= f(3)$ does not hold true. But if you take $f(x)=e^{x}$, your claim holds true. So you can't conclude monotonicity of the function from the monotonicity of its derivative.

Answer 3

Well , Rahul what you can comment about$\ f(x)$ is it's concavity . What you claim about$\ b$ is right only if in the interval$\ [2,3]$ ,$\ f'(x) \ge0$. Otherwise not .

Hint: To understand the relation between functions and monotonicity of its derivative , take few coordinates of your choice and draw arbitrary graphs of$\ f(x)$ satisfying monotonicity of$\ f'(x)$ .