`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]`

Explanation

since `f` is continuous at `x=frac{pi}{4}`


`therefore f(frac{pi}{4})-f_{h -> 0}(frac{pi}{4}+h)-f_{h -> 0}(frac{pi}{4}-h)`


`Rightarrow frac{pi}{4} cot frac{pi}{4}+b-f_{h -> 0}(frac{pi}{4}+h)+a^{2} sqrt{2} sin (frac{pi}{4}+h)`


`Rightarrow frac{pi}{4}(1)+b-(frac{pi}{4}+0)+a^{2} sqrt{2} sin (frac{pi}{4}+0)`


`Rightarrow frac{pi}{4}+b-frac{pi}{4}+a^{2} sqrt{2} sin frac{pi}{4}`


`Rightarrow b-a^{2} sqrt{2} frac{1}{sqrt{2}} Rightarrow b-a^{2}`


Also as `f` is continuous at `x-frac{pi}{4}`



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