CUET Preparation Today
The function $f:R →R$ (where $R$ is set of real numbers) defined as $f(x) = x^2 + 2x$ is |
decreasing in $(-∞, -1]$ increasing in $(-∞, -1]$ decreasing in $(-∞,2]$ increasing on $R$ |
decreasing in $(-∞, -1]$ |
The correct answer is Option (1) → decreasing in $(-∞, -1]$ Given function: $f(x) = x^2 + 2x$ Derivative: $f'(x) = 2x + 2$ A function $f(x)$ is strictly decreasing in an interval where its derivative is less than zero ($f'(x) < 0$). $2x + 2 < 0$
$2x < -2$
$x < -1$
Thus, the function is strictly decreasing in the open interval $(-\infty, -1)$. Since $f(x)$ is a polynomial function and is continuous at the boundary point $x = -1$, by standard convention for intervals of monotonicity, the endpoint is included in the final interval. Hence, the function $f(x)$ is decreasing in the interval $(-\infty, -1]$. |
