KVPY Sample Paper KVPY Stream-SX Model Paper-30

Let \[f:R\to R\] \[f(x)=\sqrt{4{{x}^{2}}+28x+85}+\sqrt{4{{x}^{2}}-28x+113}\] The minimum value off is-

A) 96

B) 14

C) \[96\sqrt{2}\]

D) \[14\sqrt{2}\]

Correct Answer: D

Solution :

\[f(x)=\sqrt{{{(2x+7)}^{2}}+{{6}^{2}}}+\sqrt{{{(2x-7)}^{2}}+{{8}^{2}}}\]

This is sum of distance of \[P=(2x,\,\,7)\]from \[A=(-7,\,\,1)\]and \[B=(7,15)\]

By triangle inequality the minimum occurs when P, A, B are collinear with P lying between A and B.

\[\therefore \,\,\,\,AB=\sqrt{{{14}^{2}}+{{14}^{2}}}=14\sqrt{2}\]