Approach Matters, Answer Doesn't

Algebra Level 3

$\large S=\left| \sqrt { { x }^{ 2 }+4x+5 } -\sqrt { { x }^{ 2 }+2x+5 } \right| \quad \forall x \in \mathbb R$

For $S$ as given above, find the maximum value of $S^8$ .

Notation: $| \cdot |$ denotes the absolute value function .

The answer is 16.

3 solutions

Deepanshu Gupta
Sep 24, 2014

$S=\left| \sqrt { { (x+(-2)) }^{ 2 }+{ (0-1) }^{ 2 } } -\sqrt { { (x+(-1)) }^{ 2 }+{ (0-2) }^{ 2 } } \right|$ .

Let $A(-2,1)$ , $B(-1,2)$ and $P(x,0)$ in the $x$ - $y$ plane $\implies S = | PA-PB |$ .

Now by triangle inequality difference of any side must be lesser or equal to third side.

$\displaystyle \Longrightarrow \left| PA-PB \right| \le \left| AB \right| \\ \Longrightarrow { S }_{ max }=AB=\sqrt { 2 } \\ \Longrightarrow { { (S }_{ max } })^{ 8 }=16$

The title compelled me to think this way...Nice problem

Nishant Sharma - 6 years, 8 months ago

Difference of any two sides of a triangle is never equal to third side, it is always less than the third side. Since co - ordinates of $P(x,0)$ is variable, so there is a possibility that given points become collinear for some value of $x$ , in which case equality occurs. This value of $x$ comes out to be $3$ .