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Q.
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
Straight Lines
Solution:
$y=f(x)=\sqrt{\left(x^{2}-2\right)^{2}+(x-3)^{2}}-\sqrt{x^{2}+\left(x^{2}-1\right)^{2}}$
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 $P A-P B \leq A B$ for possible positions of $P$.
Hence $f(x)]_{\max }=$ distance between $A B=\sqrt{9+1}=\sqrt{10}$