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Separate the intervals of monotonocity for the following function: (a) f(x) =-2x^(3)-9x^(2)-12x+1 (b) f(x)=x^(2)e^(-x) (c ) f(x) =sinx+cosx,x in (0,2pi) (d) f(x) =3cos^(4)x+cosx,x in (0,2 pi) (e ) f(x)=(log_(e)x)^(2)+(log_(e)x) |
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Answer» `f(x)=-6X^(2)-18x-12` `=-6(x+2)(x+1)` or `f'(x)gt0 if x in (-2,-1)` and `f'(x)lt0, if x in (-oo.-2)cup(-1,oo)` Thus f(x) is increasing for x `in` (-2,1) and f(X)is decreasing for x `in (-oo,2)cup(-1,oo)` (b) Let y=f(x)=`x^(2)e^(-x)` `therefore (dy)/(dx)=2xe^(-x)-x^(2)e^(-x)` `=e^(-x)(2x-x^(2))` `=e^(-x)x(2-x)` f(x) is increasing if `f(x)gt0 or x (2-x)gt0 or x in (0,2)` f(x) is decreasing if `f'(x)lt0 or x(2-x)lt0` or `x in (-oo,0)cup(2,oo)` (c ) we have f'(x) =cosx -sin x  f(x) is increasing if `f'(x) gt0 or cos x gt sin x ` or `x in (0,pi//4)cup ((5pi)/(4),2PI)` , (see the graph) f(x) is decreasing if `f'(x) lt 0 or cos x lt sinx ` or`x in (pi//4,(5pi)/(4))` (d) Given f(x) =`3cos^(4)x+10 cos^(3)x+6cos^(2)x-3` `therefore f'(x) =12 cos^(3)x(-sin x)+30 cos^(2)x(-sin x) + 12 cosx-(-sinx)` `=-3 sin 2x(2 cos^(2) x+5 cos x +2)` `=-3 sin 2x(2 cos x+1)(cos x+2)` `f'(x) =0 rarrr sin 2x =0 rarr 0,(pi)/(2), pi` or `2cos x +1 =0 rarr x=(2pi)/(3)` as `cos x+ 2 ne 0` sign scheme of f'(x) is as follow So , f (x) DECREASE on `(0,(pi)/(2))cup((2pi)/(3),pi)` and incerase on `((pi)/(2),(2pi)/(3))` (e ) f(X) =`(log_(e)x)^(2)+(log_(e)x),xgt0` `therefore f'(x) =2(log_(e)x)/(x)+(1)/(x)=(2log_(e)x+1)/(x)` f(X) increasing when 2 `log_(e)x+1gt0` or `log_(e)xgt-(1)/(2)` or `xgte^(-1)/(2)` or f(x) increases when`x in ((1)/sqrt(e),oo)` or f(x) decreases when x `in (0,(1)/sqrt(e))` |
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