# Maximum and Minimum Value of The product of two functions

Let's say I have the function g(x) which is increasing in the closed interval a<=x<=b and g(x) is always positive. And another function h(x) which is decreasing in the closed interval a<=x<=b and h(x) is always positive. Now let's define a new function i(x)= g(x) h(x) which is the product of two functions and c<=i(x)<=d . Can we define the minimum and maximum value of i(x) which is c and d as the function of a and b? For example c = g(a) h(a) . I tried to find the derivative using the product rule and found that i'(x)=g'(x)h(x)+g(x)h'(x) . However, this does not give any information whether the derivative is greater or lesser than 0 so I cannot tell whether this function is increasing or decreasing and cannot find the minimum and maximum value c and d. Is there another approach to this simple problem? Am I missing something in the process?

• You are not missing anything. $I'$ should give you information about the max and the min of $gh$. Of course that depends on the concrete functions $f$ and $g$. There is no generic answer. – GReyes Jan 31 '20 at 16:46

I think the answer is no. Here are some examples where the behavior of $$g(x)h(x)$$ can vary:
Let's just fix $$a=1, b=2$$.
I. $$g(x)=x, \quad h(x)=\frac{1}{x}$$. The product is constant.
II. $$g(x)=x, \quad h(x)=\frac{1}{x^2}$$. The product is decreasing.
III. $$g(x)=x^2, \quad h(x) = \frac{1}{x}$$. The product is increasing.