Question

If the largest positive value of the function $f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}$ is $\sqrt{k}$ where $k\in N$, then find the value of $k$.

Answer 1

Answer: 12

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 $x=6$.