Let`f'(sin x)lt0 and f''(sin x) gt0 forall x in (0,(pi)/(2))` and g(x) =f(sinx)+f(cosx)
If x = 3 is the only point of minima in its neighborhood and x=4 is neither a point of maxima nor a point minima, then which of the following can be true?

Options

A. `a lt 0, b gt 0`
B. ` a hy 0, b lt 0`
C. ` a ht 0, b lt 0`
D. not possible

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