Q. 11 - Calculus

Question

Restate Rolle’s Theorem so that its conclusion has to do with tangent lines.

Step-by-Step Solution

Answer

The restated Rolle's theorem in terms of tangent lines is:

"If $f$ is a function continuous on $[a,b]$ and differentiable on $(a,b)$ and if $f\left(a\right) = f\left(b\right) = 0$, then there exist at least one point $c\in (a,b)$ such that the tangent line drawn at the point $(c,f(c\left)\right)$ on the curve, is parallel to the $x$ axis"

Step 1. Given information.

Actual definition of Rolle's theorem:

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ and if $f\left(a\right) = f\left(b\right) = 0$, then there exists at least one value $c\in (a,b)$ for which $f\text{'}\left(c\right)=0$. 

Step 2. Remember the geometrical meaning of derivative.

Geometrically, first derivative $f\text{'}\left(x\right)$ of a function $f$ at a point $x$ means the slope of the tangent drawn at that point $(x,f(x\left)\right)$.

we have, for some point $c\in (a,b)$ the first derivative $f\text{'}\left(c\right) = 0$

$\Rightarrow$slope of the tangent line drawn at $c$ is $0$.

$\Rightarrow$ tangent line will be a horizontal line.

$\Rightarrow$ since $x$ axis is also a horizontal line, the tangent line will be parallel to the $x$ axis.