This is one of those methods where the use case is so small that they have to be very particular in the way they ask the question.

Often times they will just flat out tell you to “Use the Max-Min Inequality.”

When they use that language, they will very often be asking you to, “find upper and lower bounds,” for a given definite integral. This language is often times confusing because you are working with a definite integral that already has some given upper and lower bounds. In this situation they are referring to the bounds for the heights of your largest and smallest area rectangles.

The term “upper bound” in Max-Min Inequality problems is really asking you to find the Maximum*
y-value
*, ${\textcolor[rgb]{}{f}}_{\textcolor[rgb]{}{\mathit{max}}}$, of given equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$. The term “lower bound” in these problems is really asking you to find the Minimum *
y-value
*, ${\textcolor[rgb]{}{f}}_{\textcolor[rgb]{}{\mathit{min}}}$, of the given equation.

If you are just asked to show that an *
answer
* to a definite integral problem *
“cannot be”
* a certain value, or that the answer to a definite integral problem must “lie between” some given values.

Your first reaction will be to try and just solve the definite integral, but you will quickly see that you are not able to actually evaluate the definite integral by hand. It is then that you must remember that they *
do not care about
* the *
actual
* value of the definite integral. They only care that you show them it could not possible be the value they have given you, or that it must be inside a certain range of *
y-values
*.

If you are being asked to find the bounds, [*
a
*,*
b
*], on a definite integral, $\underset{\textcolor[rgb]{}{a}}{\overset{\textcolor[rgb]{}{b}}{\int}}\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}\textcolor[rgb]{}{\mathit{dx}}$, that would maximize or minimize the integral.

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