GATE (TF) Textile 2017 Question Paper Solution | GATE/2017/TF/30

Question 30 (Textile Engineering & Fibre Science)

The function $f(x) = x^3-3x^2-9x+10$ is

(A)	Increasing in the interval $(1,\infty)$
(B)	Increasing in the interval $(-\infty , 1)$
(C)	Decreasing in the interval $(-1,3)$
(D)	Decreasing in the interval $(-3,1)$

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$f(x) = x^3-3x^2-9x+10$
$\frac{df}{dx}=3x^2-6x-9$
For maxima ,minima-
$\frac{df}{dx}=0$
$3x^2-6x-9$ =0
$x^2-3x-3=0$
(x-3)(x+1)=0
x=-1 and 3
and
$\frac{d^2f}{dx^2}=6x-6$
$\frac{d^2f}{dx^2}=6\times -1-6=-12<0$ at x=-1, this is condition of maxima
$\frac{d^2f}{dx^2}=6\times 3-6=12>0$ at x=3, this is condition of minima
So the function is decreasing in the interval (-1, 3) (Ans)

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