The values of a for which the function f(x) = | vedclass

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(D) Given $f(x) = (a + 2)x^3 - 3ax^2 + 9ax - 1$. For $f(x)$ to decrease monotonically for all $x \in \mathbb{R}$,we must have $f'(x) \le 0$ for all $x

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$-\infty < a \le -3$

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(D) Given $f(x) = (a + 2)x^3 - 3ax^2 + 9ax - 1$. For $f(x)$ to decrease monotonically for all $x \in \mathbb{R}$,we must have $f'(x) \le 0$ for all $x \in \mathbb{R}$. $f'(x) = 3(a + 2)x^2 - 6ax + 9a \le 0$. Dividing by $3$,we get $(a + 2)x^2 - 2ax + 3a \le 0$ for all $x \in \mathbb{R}$. For a quadratic expression $Ax^2 + Bx + C \le 0$ to hold for all $x$,we require $A Here,$A = a + 2 Discriminant $D = (-2a)^2 - 4(a + 2)(3a) = 4a^2 - 12a^2 - 24a = -8a^2 - 24a$. Setting $D \le 0 \implies -8a(a + 3) \le 0 \implies a(a + 3) \ge 0$. This inequality holds when $a \le -3$ or $a \ge 0$. Combining $a Thus,the range is $(-\infty, -3]$.

The values of $a$ for which the function $f(x) = (a + 2)x^3 - 3ax^2 + 9ax - 1$ decreases monotonically for all real $x$ are:

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