If, , then
has a local maximum at
is decreasing on
there exists some such that
has a local minimum at
has a local minimum at
Explanation for the correct options:-
Step 1: Apply Leibnitz rule
Given:-
,
Differentiating both sides with respect to
We know that if the functions and are defined on and differentiable at a point , and is continuous on , then
By applying the above Leibnitz rule, we get
Step 2: Find the critical points
For critical point
are the critical points of the function
Step 3: Check the interval of monotonicity
If , because is always positive and for and
So, is increasing if .
If , because is always positive and for and .
So, is decreasing if .
If , because is always positive and for and
So, is increasing if .
Thus, the function has a local maximum at and has a local minimum at .
Step 4: Apply Rolle's theorem
Now,
So, for and
So, by Rolle's theorem, there exists some such that
As
Therefore, there exists some such that
But in option C it is given that there exists some such that , which is not correct because there is no interval which is subset of at which the value of the function is equal.
Hence, the correct options are A, B, and D.