Question

Use Lagrange's Mean Value Theorem to determine a point on the curve $y=\sqrt{x-2}$ at the tangent is parallel to the chord joining the points $(2,0)$ and $(3,1)$.

Answer 1

By LMVT, slope of tangent at $x\in (2,3)wouldbe$\frac {f(b)-f(a)}{b-a}=\frac {1-0}{3-2}=\frac {1}{1}=1$

Slope of tangent $=\frac{dy}{dx}=\frac{1}{2\sqrt{x-2}}=1$

$\Rightarrow \sqrt{x-2}=\frac{1}{2}\Rightarrow x-2=\frac{1}{4}\Rightarrow x=\frac{9}{4}$

As $\frac{9}{4}\in (2,3)\Rightarrow LMVTsatisfied$