Question

Restate the Mean Value Theorem so that its conclusion has to do with tangent lines.

Short answer

The restated theorem in terms of tangent lines is:

"If $f$is continuous on $[a,b]$and differentiable on $(a,b)$, then there exists at least one value $c\in (a,b)$such that, the tangent line of the curve drawn at $(c,f(c\left)\right)$ is parallel to the line joining the points$(a,f(a\left)\right)$and$(b,f(b\left)\right)$"

Step-by-step solution

Step 1. Given information.

The actual Mean value theorem is:

If $f$is continuous on $[a,b]$and differentiable on $(a,b)$, then there exists at least one value $c\in (a,b)$ such that,$f\text{'}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

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 $f\text{'}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$\Rightarrow$slope of the tangent line drawn at $c$is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$.

$\Rightarrow$slope of the line joining the points $(a,f(a\left)\right)$and $(b,f(b\left)\right)$is also $\frac{f\left(b\right)-f\left(a\right)}{b-a}$.

$\Rightarrow$we have "if the slopes of two lines are equal, then those lines are parallel"

that is, the tangent line at $c$and the line joining the points $(a,f(a\left)\right)$and $(b,f(b\left)\right)$are parallel.