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Q.
If the largest positive value of the function $f(x)=\sqrt{8 x-x^2}-\sqrt{14 x-x^2-48}$ is $\sqrt{k}$ where $k \in N$, then find the value of $k$.
Application of Derivatives
Solution:
We have $f(x)=\sqrt{16-(x-4)^2}-\sqrt{1-\left(x -7\right)^2}$
$\text { Now consider } y=\sqrt{16-(x-4)^2} $
$\Rightarrow (x-4)^2+y^2=16, y>0$
is a semi circle with cenre $(4,0)$ and radius $=4$.
$\text { Hy } y=\sqrt{1-(x-7)^2} $
$\Rightarrow (x-7)^2+y^2=1, y>0$
is a semi circle with centre $(7,0)$ and radius $=1$
Now on combining the 2 figures, we have
$f(x)_{\max }=$ maximum vertical distance between the 2 curves which occurs when $x=6$.
