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 $(5,0)$

Tangent parallel to the chord joining $A(4,1),B(6,1)$ will touch the curve at point $(c,f(c\left)\right)$ as mean value theorem is applied on $[4,6]$

$\Rightarrow f^{\prime }\left(c\right)=\frac{f\left(6\right)-f\left(4\right)}{6-4},c\in (4,6)$
$\Rightarrow 2(c-5)=\frac{1-1}{2}$
$\Rightarrow 2(c-5)=0$
$\Rightarrow c=5\Rightarrow f\left(c\right)=0$
Required point is $(c,f(c\left)\right)\equiv (5,0)$