Monotonicity, Maxima and Minima
1.0 Rolle's Theorem
1.0 Rolle's Theorem
Statement: Let $f$ be any continuous function on the closed interval $[a,b]$ and also differentiable on the open interval $(a,b)$ and also satisfying the condition $$f(a)=f(b)$$Then there exists atleast one real number $'c'$ lies between $'a'$ 
and $'b'$ such that $$f'(c)=0$$
and $'b'$ such that $$f'(c)=0$$To understand the statement of the theorem, let us take one of the condition on the function $f(x)$ i.e., continuous and differentiable function in the particular interval. There are only two possible situations on the function $f(x)$ when it is continuous and differentiable in any interval.
Situation 1. $f(x)$ is a constant function in the interval $[a,b]$.
Let $f(x)=c$ be any constant function as shown in figure, then we can say that $$f'(x)=0$$ for all values of $x$ lies between $a$ and $b$ which proves the statement of Rolle's theorem that $$f'(c)=0$$ such that $a<c<b$.
Situation 2. $f(x)$ is not constant function in the interval $[a,b]$. Since $f(a)=f(b)$, the function must be first increasing then decreasing or vice-versa in the interval $[a,b]$ as shown in figure $1$.
Let us assume the point at which the maximum occurs be $C$ at $x=c$. Now, we can easily understand why in the statement of Rolle's theorem, there is a condition that the function must be differentiable in $(a.b)$. There is a possibility that the function is satisfying all the other conditions i.e., $f(a)$$=$$f(b)$ but at point $c$, we can not find $f'(c)$ as shown in figure $2$. In such case the rolle's theorem is not applicable.
If the function is continuous and differentiable and $f(a)=f(b)$, then there will always be a point $c$ between $a$ and $b$ such that at point $c$, we can draw a horizontal tangent i.e., $f'(c)=0$ which proves the statement of the theorem.
