1.

`f(x)={x+sqrt2asinx,0ltxltpi/4 and 2xcotx+b , pi/4lexltpi/2 and acosx-bsinx , pi/2lexltpi `. Determine the value of a and b If function is continuous for interval `[0,pi]`

Answer» Here, value of `f(x)` is changing at `pi/2` and `pi/4`.
So, `f(x)` to be continuous,
`f((pi/4)^-) = f((pi/4)^+)`
`=>pi/4+sqrt2 a*1/sqrt2 = 2*pi/4(1)+b`
`=>pi/4 +a = pi/2+b`
`=>a-b = pi/4->(1)`
Also, `f(x)` to be continuous,
`f((pi/2)^-) = f((pi/2)^+)`
`=>2(0) + b = a(0) - b(1)`
`=>2b = 0`
`=> b = 0`
Putting `b = 0` in (1),
`=> a - 0 = pi/4`
`=> a = pi/4`
So, for `a = pi/4` and `b = 0`, `f(x)` will be continuous.


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