Question

select the correct answer.
which statement is true about this radical function?
f(x) = 3\sqrt3{-x} + 2
a. the function is strictly increasing.
b. the function is strictly decreasing.
c. the function is increasing and decreasing.
d. the function is constant.

Explanation:

Step1: Rewrite the function

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

Step2: Find the derivative

Use power rule: $\frac{d}{dx}[x^n]=nx^{n-1}$
$f'(x) = 3 \cdot \frac{1}{3}(-x)^{-\frac{2}{3}} \cdot (-1) = -\frac{1}{(-x)^{\frac{2}{3}}} = -\frac{1}{\sqrt[3]{x^2}}$

Step3: Analyze the derivative

For all $x
eq 0$, $\sqrt[3]{x^2} > 0$, so $f'(x) = -\frac{1}{\sqrt[3]{x^2}} < 0$. At $x=0$, the function is still decreasing as the left and right behavior confirms a negative slope.

Answer:

B. The function is strictly decreasing.