1.

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

Answer»

`a lt 0, b gt 0`
` a hy 0, b lt 0`
` a ht 0, b lt 0`
not possible

Solution :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


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