Q.Using the Rolle's theorem, determine the values of $x$ at which the tangent is parallel to the $x$-axis for the following functions:

Both theorems below guarantee the existence of a point where the tangent behaves in a special way — they do not by themselves tell you how many such points there are, or give a formula beyond the equation used to solve for them.

Intermediate Value Theorem. If $f$ is continuous on $[a,b]$, then $f$ takes every value between $f(a)$ and $f(b)$ somewhere in $[a,b]$.

Rolle's Theorem. If $f(x)$ is continuous on the closed interval $[a,b]$, differentiable on the open interval $(a,b)$, and $f(a)=f(b)$, then there is at least one $c\in(a,b)$ with $f'(c)=0$.

Geometric meaning: if the endpoint heights match, somewhere in between the tangent must be horizontal (parallel to the $x$-axis).

Rolle's theorem can also be used indirectly, without solving an equation, to bound how many real roots an equation can have in an interval: if $f$ had two roots $\alpha<\beta$ in $(a,b)$ with $f$ continuous/differentiable there, Rolle's theorem would force a zero of $f'$ strictly between them — so if $f'(x)\ne0$ throughout $(a,b)$, $f$ can have at most one root there.

Failure modes for Rolle's theorem (why it does not apply): continuity fails on $[a,b]$ (e.g. an undefined or infinite point inside the interval), differentiability fails somewhere in $(a,b)$ (e.g. a corner, as in $|x|$-type functions), or simply $f(a)\ne f(b)$.

Lagrange's Mean Value Theorem (LMVT). If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$ (with $f(a),f(b)$ not necessarily equal), then there is at least one $c\in(a,b)$ with

Rolle's theorem is the special case $f(a)=f(b)$ (LMVT with right side $=0$) — it is sometimes called the "rotated Rolle's theorem."

Geometric meaning: the tangent at some interior point is parallel to the secant joining the two endpoints — equivalently, the instantaneous rate of change equals the average rate of change over $[a,b]$ at some interior instant.

Consequences of LMVT (used throughout the rest of the chapter).

1. If $f'(x)>0$ for all $x\in(a,b)$, then $f$ is strictly increasing on $(a,b)$; if $f'(x)<0$ throughout, $f$ is strictly decreasing (this is exactly Theorem 7.7/7.4's monotonicity test).
2. If $f'(x)=0$ for all $x\in(a,b)$, then $f$ is constant on $(a,b)$.
3. If $f'(x)=g'(x)$ for all $x$, then $f(x)=g(x)+C$ for some constant $C$.

Typical uses. Beyond finding the guaranteed $c$ directly, LMVT is a standard tool for proving an inequality — bound $f'(c)$ using a given bound on $f'$, then substitute into $f'(c)=\dfrac{f(b)-f(a)}{b-a}$ to bound $f(b)-f(a)$ itself; or apply it to $f(x)=e^{-x}$, $f(x)=\sin x$, etc., between two arbitrary points to derive a general inequality valid for all values in a domain.

Failure modes for LMVT: the same two culprits as Rolle's — a break in continuity anywhere on $[a,b]$ (commonly, an undefined point) or a break in differentiability somewhere in $(a,b)$ (commonly, a corner from an absolute value).