Question

algebra 2 362403-6 fy - i
benchmark test 2
over what interval is the graph of the polynomial function $f(x) = x^3 - 6x^2 + 3x + 10$ increasing?
a. $(-infty, 0.27)$ and $(3.73, infty)$
b. $(-1, 2)$ and $(5, infty)$
c. $(-infty, -1)$ and $(2, 5)$
d. $(5, infty)$ and $(2, 5)$

Explanation:

Step1: Find derivative of $f(x)$

$f'(x) = 3x^2 - 12x + 3$

Step2: Solve $f'(x)=0$ for critical points

Use quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
For $3x^2-12x+3=0$, $a=3, b=-12, c=3$:

$$\begin{align*} x&=\frac{12\pm\sqrt{(-12)^2-4(3)(3)}}{2(3)}\\ &=\frac{12\pm\sqrt{144-36}}{6}\\ &=\frac{12\pm\sqrt{108}}{6}\\ &=\frac{12\pm6\sqrt{3}}{6}\\ &=2\pm\sqrt{3} \end{align*}$$

Step3: Approximate critical points

$2+\sqrt{3}\approx3.73$, $2-\sqrt{3}\approx0.27$

Step4: Test sign of $f'(x)$

• For $x<0.27$, $f'(x)>0$ (function increasing)
• For $0.27
• For $x>3.73$, $f'(x)>0$ (function increasing)

Answer:

A. $(-\infty, 0.27)$ and $(3.73, \infty)$