Mean Value Theorem: Showing Change in a Function is Bounded

In summary, the Mean Value Theorem is a mathematical concept that states for a continuous and differentiable function, there exists at least one point in the interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints. It is used to show that the change in a function over a given interval is bounded and can only be applied to continuous and differentiable functions. The Mean Value Theorem can also be used to prove the existence of extrema, but it cannot provide the exact value of the maximum or minimum. Other methods are needed for this.
  • #1
karush
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MHB
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Ok Just have trouble getting this without a function..
 
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  • #2
average rate of change of $f(x)$ on the interval $[2,8]$ is $\dfrac{f(8)-f(2)}{8-2}$

the MVT states there exists at least one value of $x \in (2,8)$ where $f'(x) = \dfrac{f(8)-f(2)}{8-2}$

$3 \le f'(x) \le 5 \implies 3 \le \dfrac{f(8)-f(2)}{8-2}\le 5 \implies 18 \le f(8)-f(2) \le 30$
 

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