Find the intervals in which `f(x)` is increasing or decreasing:
(1). `f(x)=x|x|, x in R`
(2). `f(x)=sinx+|sinx|, 0 < x le 2pi`

Explanation

`f(x)=sin x+|sin x|, 0

Case I: `When x in(0, pi) `
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f(x)=sin x+sin x=2 sin x `
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Rightarrow f^{prime}(x)=2 cos x`
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As, cos x>0 for x in(0, frac{pi}{2}) and cos x<0 for x in(frac{pi}{2}, pi) `
` f^{prime}(x)>0 for x in(0, frac{pi}{2}) and f^{prime}(x)<0 for x in(frac{pi}{2}, pi) `
therefore `f(x)` is increaing on (0, `frac{pi}{2}) and f(x) `is decreasing on` (frac{pi}{2}, pi). `
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