Let `f(x)=xsqrt(4a x-x^2),(a >0)dot` Then `f(x)` is (a). increasing in (0,3a) decreasing in (3a, 4a) (b). increasing in (a, 4a) decreasing in`(5a ,oo)` (c). increasing in `(0,4a)` (d). none of these

Options

A. increasing in (0,3a) decreasing in (3a,4a)
B. increasing in (a,4a),decreasing in `(5a,oo)`
C. increasing in `(0,4a)`
D. none of these

Answer

1

Explanation

f(X)=`xsqrt(4ax-x^(2))`
`therefore f(X) =sqrt(4ax-x^(2))+x(4a-2x)/2sqrt(4ax-x^(2))`
`=(2x(3a-x))/sqrt(4ax-x^(2))`
Now if `f(X)gt0` then
`2x(3a-x)gt0`
`2x(x-3a)lt0`
or x in (0,3a)
Thus f(x) increases in (0,3a) and decreases in (3a,4a)
Thus f(X) increases in (0,3a) and decreases (3a,4a)