Let`f'(sin x)lt0 and f''(sin x) gt0 forall x in (0,(pi)/(2))` and g(x) =f(sinx)+f(cosx)
consider function f(x)=`{{:(-x^(2)+4x+a,xle3),(ax+b,3ltxlt4),(-b/4x+6,xge4):}`
which of the following is true?

Options

A. f(x) is discounting function for all ordered pairs (a,b)
B. f(x) is contionuous for finite number of ordered pairs (,ab)
C. f(x) can be differentiable
D. f(x) is continous for infinite ordered pairs (a,b)

Answer

4

Explanation

If f(X) is continous then `f(3^(-))=f(3^(+))`
or -9+12+a=3a+b or 2a+b=3
Also `f(4^(+)) or 4a+b=-b+6 r 2a+b=3`
Thus f(x) is contnous for infinite values of a and b also
`f(x)={{:(-2x+4,xlt3),(a,3ltxlt4),((-b)/(4),xgt4):}`
For f(x) to be diffentiable
`f(3^(-))=f(3^(+))`
or `a=-2 and -(bb)/(4) =a=-2 or b=8`
But these values do not satisfy equation (1)
Hence f(x) cannot be differentiable