Question

the function f is given by f(x)=x² + 3x - 6. which of the following describes f? a for any interval of x, the function always has a positive rate of change. b for any interval of x, the function always has a negative rate of change. c for any interval of x < -1.5, the function has a positive rate of change, and for any interval of x > -1.5, the function has a negative rate of change. d for any interval of x < -1.5, the function has a negative rate of change, and for any interval of x > -1.5, the function has a positive rate of change.

Explanation:

Step1: Find the derivative of the function

Given $f(x)=x^{2}+3x - 6$, using the power - rule $(x^n)'=nx^{n - 1}$, we have $f'(x)=2x + 3$.

Step2: Find the critical point

Set $f'(x)=0$, so $2x+3 = 0$. Solving for $x$, we get $x=-\frac{3}{2}=-1.5$.

Step3: Analyze the sign of the derivative

When $x\lt - 1.5$, let's take $x=-2$, then $f'(-2)=2\times(-2)+3=-1\lt0$, so the function has a negative rate of change.
When $x\gt - 1.5$, let's take $x = 0$, then $f'(0)=2\times0 + 3=3\gt0$, so the function has a positive rate of change.

Answer:

D. For any interval of $x\lt - 1.5$, the function has a negative rate of change, and for any interval of $x\gt - 1.5$, the function has a positive rate of change.