Question

Is the Mean Value Theorem same as the Intermediate Value Theorem?

Answer 1

Hint: You need to keep proper concepts on the statement of the theorems “Mean Value Theorem” and “Intermediate Value Theorem”. Compare the statements of these two theorems.

Complete step by step solution:
The statement of the “Mean Value Theorem” is
If $f\left( x \right)$ be a function defined on the interval $\left[ {a,b} \right]$such that

• • $f\left( x \right)$ is continuous in the interval $\left[ {a,b} \right]$

• $f\left( x \right)$ is differentiable in the interval $\left( {a,b} \right)$

Then there exists a value of $x$, say $x = c$, which lies in the interval $\left( {a,b} \right)$ such that $\dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}} = f'\left( c \right)$.
The statement of the “Intermediate Value Theorem” is
If $f\left( x \right)$ be a continuous function defined in the interval $\left[ {a,b} \right]$ and $k$ is a value between $f\left( a \right)$ and $f\left( b \right)$. Then there exists a value of $x$, say $x = c$, which lies in the interval $\left( {a,b} \right)$ such that $f\left( c \right) = k$.
From the statements of the two theorems, it is clear that the Mean Value Theorem is not the same as the Intermediate Value Theorem.
The condition of existence of differentiability is in the Mean Value Theorem but not in the Intermediate Value Theorem.

Note: Do not get confused with Mean value theorem and Intermediate value theorem. Remember that the mean value theorem includes both continuity and differentiability but the Intermediate value theorem includes continuity only.