Question

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

• A. $1<\mathrm{x}<3$
• B. $\mathrm{x}<0$
• C. $\mathrm{x}>0$
• D. $0<\mathrm{x}<3/2$

Answer 1

The correct option is A

$1<\mathrm{x}<3$

Explanation for the correct option:

Option (A): Given that $\mathrm{y}=\mathrm{x}{(\mathrm{x}-3)}^2$

Graphical representation of $\mathrm{y}=\mathrm{x}{(\mathrm{x}-3)}^2$ is as follows:

From the graph it is clear that the curve of $\mathrm{y}=\mathrm{x}{(\mathrm{x}-3)}^2$ decreases in the interval$1<\mathrm{x}<3$

Hence, Option (A) is correct.

Explanation for the incorrect options:

Option (B) From the graph it is clear that the curve of $\mathrm{y}=\mathrm{x}{(\mathrm{x}-3)}^2$ increases in the interval $\mathrm{x}<0$

Option (C) From the graph it is clear that the curve of $\mathrm{y}=\mathrm{x}{(\mathrm{x}-3)}^2$ increases in the interval $\mathrm{x}<1$ and then decreases in the interval $1<\mathrm{x}<3$. and again increases in the interval $3<x$ Hence $\mathrm{y}=\mathrm{x}{(\mathrm{x}-3)}^2$ does not decrease for all values of $\mathrm{x}$ in the interval $\mathrm{x}>0$.

Option (D) From the graph it is clear that the curve of $\mathrm{y}=\mathrm{x}{(\mathrm{x}-3)}^2$ increases in the interval $0<\mathrm{x}<1$ and then decreases in the interval $1<\mathrm{x}<\frac{3}{2}$. Hence $\mathrm{y}=\mathrm{x}{(\mathrm{x}-3)}^2$ does not decrease for all values of $\mathrm{x}$in the interval $0<\mathrm{x}<\frac{3}{2}$.

Therefore, option (B), (C) and (D) are incorrect.

Hence the correct option is Option (A).