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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?

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  • $\begingroup$ 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. $\endgroup$ – GReyes Jan 31 '20 at 16:46
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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.

You can come up with other examples. The point is that with the information provided, we cannot say whether the product of the functions increases or decreases on the interval, and where the minimum and maximum are (without knowing more about the functions).

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