Question

given the function $y = 3x^4 - 8x^3$, select all of the statements that are true about the graph of the function.
the function has zeroes at 0 and 2.67.
the domain of the function is all real numbers.
the function has a minimum value of -16.
the function has negative values on the intervals: $(-\infty, 0)$ and $(2.67, \infty)$.
the end behavior of the function is: as $x \to -\infty$, $f(x) \to \infty$, and as $x \to \infty$, $f(x) \to -\infty$.
the function is increasing on the intervals: $(-\infty, 0)$ and $(2.67, \infty)$.
the function has no symmetry.

Explanation:

Step1: Find zeros of the function

Set $y=0$:
$$3x^4 - 8x^3 = 0$$
$$x^3(3x - 8) = 0$$
Solutions: $x=0$ and $x=\frac{8}{3}\approx2.67$

Step2: Determine the domain

Polynomial functions have a domain of all real numbers, since there are no restrictions like division by zero or square roots of negative numbers.

Step3: Find critical points and minimum

Take the first derivative:
$$y' = 12x^3 - 24x^2 = 12x^2(x - 2)$$
Set $y'=0$: critical points at $x=0$ and $x=2$.
Evaluate $y$ at $x=2$:
$$y=3(2)^4 - 8(2)^3 = 48 - 64 = -16$$
Second derivative test: $y''=36x^2-48x$, $y''(2)=36(4)-48(2)=144-96=48>0$, so $x=2$ is a local minimum. As $x\to\pm\infty$, $y\to+\infty$, so this is the global minimum.

Step4: Analyze sign of the function

Test intervals:

• For $x<0$: $x^3<0$, $3x-8<0$, so $y=x^3(3x-8)>0$
• For $00$, $3x-8<0$, so $y<0$
• For $x>\frac{8}{3}$: $x^3>0$, $3x-8>0$, so $y>0$

Step5: Determine end behavior

The leading term is $3x^4$, degree 4 (even) with positive coefficient. So as $x\to-\infty$, $y\to+\infty$; as $x\to+\infty$, $y\to+\infty$.

Step6: Find increasing/decreasing intervals

Use $y'=12x^2(x-2)$:

• For $x<2$ (excluding $x=0$, where $y'=0$ but no sign change): $y'\leq0$, function is decreasing
• For $x>2$: $y'>0$, function is increasing

Step7: Check symmetry

Test for even function: $y(-x)=3(-x)^4-8(-x)^3=3x^4+8x^3
eq y(x)$
Test for odd function: $y(-x)
eq -y(x)=-3x^4+8x^3$, so no symmetry.

Answer:

• The function has zeroes at 0 and 2.67.
• The domain of the function is all real numbers.
• The function has a minimum value of -16.
• The function has no symmetry.