If y = x ( x - 3 ) 2 decreases for the values of x given by:

Question

If $y = x {(x - 3)}^{2}$ decreases for the values of $x$ given by:

Smart Achievers · Accepted Answer

(1)

Ans. $1 < x < 3$

Explanation:

We have, $y = x {(x - 3)}^{2}$

$∴$ $\frac{d y}{d x} = x \cdot 2 (x - 3) \cdot 1 + {(x - 3)}^{2} \cdot 1$

$= 2 x^{2} - 6 x + x^{2} + 9 - 6 x$

$= 3 x^{2} - 12 x + 9$

$= 3 (x^{2} - 3 x - x + 3) = 3 (x - 3) (x - 1)$

So, $y = x {(x - 3)}^{2}$ decreases for (1, 3).

[Since $y' < 0$ for all $x \in (1, 3)$ , hence $y$ is decreasing on (1, 3)]

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