Question

For the curve $f\left(x\right)=(x-5{)}^2$, applying mean value theorem on $[4,6]$ the tangent at _____ is parallel to the chord joining $A(4,1),B(6,1)$.

• A. $(4,6)$
• B. $(\frac{9}{2},\frac{1}{4})$
• C. $(0,5)$
• D. $(5,0)$

Answer 1

The correct option is D $(5,0)$

Given curve is $f\left(x\right)=(x-5{)}^2$

$f^{\prime }\left(x\right)=2(x-5)$

Now, $f$ is continuous and differential on $(4,6)$

So, applying Mean value theorem on $[4,6]$ we get

$f^{\prime }\left(c\right)=2(c-5)=\frac{f\left(6\right)-f\left(4\right)}{6-4}=0$
$\Rightarrow 2(c-5)=0$
$\therefore c=5$

At $c=5,f\left(c\right)=0$
$\therefore (5,0)$.