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Home  >>  CBSE XII  >>  Math  >>  Application of Derivatives
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At what point,the slope of the curve $y=-x^3+3x^2+9x-27$ is maximum?Also find the maximum slope.

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  • Let $f(x)$ be a function with domain $D\: \subset \: R$. Then $f(x)$ is said to attain the maximum value at a point, $ a \in D,$ if $ f'(x) \leq f(a)$ for all $ x \in D$.
Step 1
Differentiating w.r.t $x$ we get
If $f'(x)=0$
$ \Rightarrow -3x^2+6x+9=0$
$ 3(-x^2+2x+3)=0$
factorising this we get,
$ f'(x)=(x+3)(x-1)$
when $x=-3$
$ = -27+27+27-27$
when $ x=+1$
$ = -16$
Hence the point is $(1, -16)$
The maximum slope =
$ f'(1)=-3(1)+6(1)+9$
$ = 12$
answered Aug 6, 2013 by thanvigandhi_1

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