Let 𝒇(𝒙) = 𝒙² − 𝟐𝒙 + 𝟐 be a continuous function defined on 𝒙 ∈ [𝟏, 𝟑]. The point 𝒙 at which the tangent of 𝒇(𝒙) becomes parallel to the straight line joining 𝒇(𝟏) and 𝒇(𝟑) is

Answer :

Solution :

f(x) = x2 – 2x + 2, x ϵ [1, 3]

By lagrangian mean value theorem

$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$f^{\prime }\left(c\right)=\frac{f\left(3\right)-f\left(1\right)}{3-1}$

$2c-2=\frac{\left(9-6+2\right)-\left(1-2+2\right)}{3-1}$

$2c-2=\frac{5-1}{2}$

c = 2