Let f(x) = {[(1-sin^3x)(3 cos^2x), if x<π/2][a,if x =π/2][b(1-sinx)/(π-2x)^2,if x>π/2]. If f(x) is continuous at π/2 find a and b. ​

Question

Let \(f(x) = \begin{cases}\frac{1-sin^3x}{3\,cos^2x}&,\quad if\,x <\frac{\pi}{2}\\a&,\quad if\, x=\frac{\pi}{2}\\\frac{b(1-sin\,x)}{(\pi-2x)^2}&;\quad if \,x>\frac{\pi}{2}\end{cases} \). If f(x) is continuous at \(\frac{\pi}{2}\) find a and b.

Answer 1

Given : 

f(x) is continuous at x = \(\frac{\pi}{2}\)& f(\(\frac{\pi}{2}\)) = a.

If f(x) to be continuous at x = \(\frac{\pi}{2}\),

We have to show,

 f(\(\frac{\pi}{2}\))^- =  f(\(\frac{\pi}{2}\))⁺ =  f(\(\frac{\pi}{2}\))

∴ cos(0) = 1

f(x) is continuous at x =\(\frac{\pi}{2}\)  & f(\(\frac{\pi}{2}\)) = a, and 

from (1) & (2),we get