Question

What is the conclusion of the Mean Value Theorem?

Answer 1

Hint: To solve this question we need to have the knowledge of Mean Value Theorem. Mean value theorem applies that for a function f is continuous on the closed interval and differentiable on the open interval then there exist a point c in the interval such that differentiation of the function at “c” is equal to the functions average rate of change of over a and b on writing it mathematically it is $\dfrac{f\left( b \right)-f\left( a \right)}{b-a}$ .

Complete step-by-step solution:
The question asks us to give the conclusion for a mean value theorem. So the discussion for this question will start from knowing about the Mean Value Theorem. The Mean Value Theorem is a condition which is applied for getting the value of “c” which is in the interval of the set of numbers.
The conclusion of mean value theorem is that if a function f is continuous on the interval $\left[ a,b \right]$ then also differentiable on the $\left( a,b \right)$ then exist a point “c” in the interval $\left( a,b \right)$ such that ${f}'\left( c \right)$ which is equal to the ratio of the difference of the function $f\left( a \right)$ and $f\left( b \right)$. “c” should be the point which lies between $a$ and $b$such that the tangent is parallel to the line which passes through the points of a$\left( a,f(a) \right)$ and $\left( b,f(b) \right)$.

Note: The ultimate value of the mean value theorem is that it forces differential equations to have solutions. It can even be used to prove that integrals exist, without using sums at all, and allows you to create estimates about the behavior of those solutions.