Step 1: Assure that the requirements of the Mean Value Theorem are met by your situation or equation.
In order to be able to apply the conclusion of the MVT, you must first show or more often just state you know that the “if” potion of the theorem has been satisfied. If the function is continuous on a closed interval [ a , b ] and differentiable on the open interval ( a , b ), then you can apply the conclusion of the MVT.
Often you will be given a function that is a polynomial. The great thing about polynomials is that they are guaranteed to be continuous and differentiable everywhere in their domain.
Keep in mind that you are never going to be able to show specifically every point in an interval are continuous and differentiable. On any interval there is an infinite number of values between the endpoints, which means stating obvious conclusions about the entire interval without hand worked proof is absolutely acceptable .
Step 2: Find the averagerate of change, slope ,between the two x-value endpoints, [ a , b ] of your function.
When you are dealing with real-world problems this answer is often times the conclusion you are trying to show.
$\textcolor[rgb]{}{\mathit{average}}\textcolor[rgb]{}{\mathit{rate\; of\; change}}\textcolor[rgb]{}{=}\frac{\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{b}\textcolor[rgb]{}{\right)}\textcolor[rgb]{}{\u2013}\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{a}\textcolor[rgb]{}{\right)}}{\textcolor[rgb]{}{b}\textcolor[rgb]{}{\u2013}\textcolor[rgb]{}{a}}$
Step 3: Find the derivative of your equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.
Step 4: Set your derivative, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, equal to your $\textcolor[rgb]{}{\mathit{average}}\textcolor[rgb]{}{\mathit{rate\; of\; change}}=\frac{\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{b}\textcolor[rgb]{}{\right)}\textcolor[rgb]{}{\u2013}\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{a}\textcolor[rgb]{}{\right)}}{\textcolor[rgb]{}{b}\textcolor[rgb]{}{\u2013}\textcolor[rgb]{}{a}}$, then solve that algebra problem for x .
The x-value(s) that you find will include the c-value that the Mean Value Theorem guaranteed you would exist. Make sure you only include x-values in your answer that are inside your open interval ( a , b ).