Math / Calculus

QuestionFind the critical points and determine if the function is increasing or decreasing on the given intervals. $y=3 x^{4}+6 x^{3}$
Left critical point: $c_{1}=$ $\square$ Right critical point: $c_{2}=$ $\square$ The function is: ? on $\left(-\infty, c_{1}\right)$ . ? on $\left(c_{1}, c_{2}\right)$ . ? on $\left(c_{2}, \infty\right)$ .

Studdy Solution

STEP 1

1. A critical point occurs where the derivative is zero or undefined.
2. The function $y = 3x^4 + 6x^3$ is a polynomial, so its derivative will be defined everywhere.
3. To determine if the function is increasing or decreasing, we will analyze the sign of the derivative on the intervals defined by the critical points.

STEP 2

1. Find the derivative of the function.
2. Solve for critical points by setting the derivative equal to zero.
3. Determine the sign of the derivative on the intervals defined by the critical points.
4. Conclude whether the function is increasing or decreasing on each interval.

STEP 3

Find the derivative of the function $y = 3x^4 + 6x^3$ .
$y' = \frac{d}{dx}(3x^4 + 6x^3)$
Using the power rule, the derivative is:
$y' = 12x^3 + 18x^2$

STEP 4

Solve for critical points by setting the derivative equal to zero:
$12x^3 + 18x^2 = 0$
Factor the expression:
$6x^2(2x + 3) = 0$
This gives us two factors to solve:
1. $6x^2 = 0$ implies $x = 0$
2. $2x + 3 = 0$ implies $x = -\frac{3}{2}$
Thus, the critical points are $c_1 = -\frac{3}{2}$ and $c_2 = 0$ .

STEP 5

Determine the sign of the derivative on the intervals defined by the critical points: $(-\infty, c_1)$ , $(c_1, c_2)$ , and $(c_2, \infty)$ .
- For $x \in (-\infty, -\frac{3}{2})$ , choose a test point like $x = -2$ : $y'(-2) = 12(-2)^3 + 18(-2)^2 = -96 + 72 = -24$ The derivative is negative, so the function is decreasing.
- For $x \in (-\frac{3}{2}, 0)$ , choose a test point like $x = -1$ : $y'(-1) = 12(-1)^3 + 18(-1)^2 = -12 + 18 = 6$ The derivative is positive, so the function is increasing.
- For $x \in (0, \infty)$ , choose a test point like $x = 1$ : $y'(1) = 12(1)^3 + 18(1)^2 = 12 + 18 = 30$ The derivative is positive, so the function is increasing.

STEP 6

Conclude whether the function is increasing or decreasing on each interval:
- The function is decreasing on $(-\infty, c_1)$ . - The function is increasing on $(c_1, c_2)$ . - The function is increasing on $(c_2, \infty)$ .
Left critical point: $c_1 = -\frac{3}{2}$ Right critical point: $c_2 = 0$
The function is: decreasing on $(-\infty, c_1)$ . increasing on $(c_1, c_2)$ . increasing on $(c_2, \infty)$ .

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