Question

The maximum value of $y=\sqrt{(x-3{)}^2+{\left(x^2-2\right)}^2}-\sqrt{x^2+{\left(x^2-1\right)}^2}$ is

• A. $3$
• B. $\sqrt{10}$
• C. $2\sqrt{5}$
• D. none of these

Answer 1

Answer: (B)

$y=f(x)=\sqrt{{\left(x^2-2\right)}^2+(x-3{)}^2}-\sqrt{x^2+{\left(x^2-1\right)}^2}$
Note that the first term describes the distance between $P\left(x,x^2\right)$ and $A(3,2)$ whereas the second term describes the distance between $P\left(x,x^2\right)$ and $B(0,1)$.
Now $PA-PB\le AB$ for possible positions of $P$.
Hence $f(x){]}_{\max }=$ distance between $AB=\sqrt{9+1}=\sqrt{10}$