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
Best Answer
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?