Increasing and Decreasing Domain of a Parabola: Representation of missing c value

To determine the intervals of increase and decrease for the function $y = 6x^2 - 15x$, we begin by finding its first derivative.

The first derivative of the function is found as follows:

$y = 6x^2 - 15x$

$y' = \frac{d}{dx}(6x^2 - 15x) = 12x - 15$

To find critical points, set the derivative equal to zero:

$12x - 15 = 0$

$12x = 15$

$x = \frac{15}{12} = \frac{5}{4} = 1\frac{1}{4}$

The critical point is $x = 1 \frac{1}{4}$. We need to determine the sign of the derivative on either side of this point to identify the intervals of increase and decrease.

• For $x < 1 \frac{1}{4}$, choose $x = 0$:

$y'(0) = 12(0) - 15 = -15$

Since $y' < 0$, the function is decreasing for $x < 1 \frac{1}{4}$.

• For $x > 1 \frac{1}{4}$, choose $x = 2$:

$y'(2) = 12(2) - 15 = 24 - 15 = 9$

Since $y' > 0$, the function is increasing for $x > 1 \frac{1}{4}$.

Therefore, the function increases for $x > 1 \frac{1}{4}$ and decreases for $x < 1 \frac{1}{4}$.

The correct intervals of increase and decrease for the function $y = 6x^2 - 15x$ are:

$\nearrow: x < 1 \frac{1}{4}$ (Increasing)
$\searrow: x > 1 \frac{1}{4}$ (Decreasing)

To determine the intervals of increase and decrease for the function $y = \frac{1}{3}x^2 + 2\frac{1}{3}x$, we will perform the following steps:

• Step 1: Differentiate the function with respect to $x$.
• Step 2: Set the derivative equal to zero to find the critical points.
• Step 3: Use sign analysis on intervals determined by the critical points to identify where the function is increasing or decreasing.

Let's proceed with the solution:

Step 1: Differentiate $y = \frac{1}{3}x^2 + 2\frac{1}{3}x$.

The derivative $f'(x)$ is given by:

$f'(x) = \frac{d}{dx}\left(\frac{1}{3}x^2\right) + \frac{d}{dx}\left(2\frac{1}{3}x\right)$.

This simplifies to:

$f'(x) = \frac{2}{3}x + 2\frac{1}{3}$.

Converting $2\frac{1}{3}$ to an improper fraction gives $\frac{7}{3}$, hence:

$f'(x) = \frac{2}{3}x + \frac{7}{3}$.

Step 2: Solve $f'(x) = 0$ to find critical points.

Set $\frac{2}{3}x + \frac{7}{3} = 0$.

Multiply through by 3 to eliminate fractions:

$2x + 7 = 0$.

This simplifies to:

$2x = -7$ $\Rightarrow x = -\frac{7}{2}$ or $x = -3.5$.

Step 3: Perform sign analysis around the critical point $x = -3.5$.

• For $x < -3.5$, choose a test point like $x = -4$.

$f'(-4) = \frac{2}{3}(-4) + \frac{7}{3} = -\frac{8}{3} + \frac{7}{3} = -\frac{1}{3}$.

This is negative, indicating the function is decreasing on this interval.

• For $x > -3.5$, choose a test point like $x = 0$.

$f'(0) = \frac{2}{3}(0) + \frac{7}{3} = \frac{7}{3}$.

This is positive, indicating the function is increasing on this interval.

Thus, the function is decreasing for $x < -3.5$ and increasing for $x > -3.5$.

Therefore, the correct answer is: $\searrow: x > -3.5$; $\nearrow: x < -3.5$.

To determine the intervals of increase and decrease for the quadratic function $y = -\frac{1}{6}x^2 + 3\frac{2}{3}x$, follow these steps:

• Step 1: Find the derivative of the function:

The function is given by $y = -\frac{1}{6}x^2 + \frac{11}{3}x$.
The derivative, using power rules, is $\frac{dy}{dx} = -\frac{1}{3}x + \frac{11}{3}$.

• Step 2: Set the derivative equal to zero and solve for $x$ (Critical points):

Set $-\frac{1}{3}x + \frac{11}{3} = 0$.
Solving for $x$, we get $\frac{1}{3}x = \frac{11}{3}$,
Thus, $x = 11$.

• Step 3: Determine the intervals by testing values around the critical point:

For $x < 11$, choose any point like $x = 0$:
$\frac{dy}{dx} = -\frac{1}{3}(0) + \frac{11}{3} = \frac{11}{3} > 0$. So, the function is increasing on $x < 11$.
For $x > 11$, choose any point like $x = 12$:
$\frac{dy}{dx} = -\frac{1}{3}(12) + \frac{11}{3} = -4 + \frac{11}{3} = -\frac{1}{3} < 0$. So, the function is decreasing on $x > 11$.

Thus, the function is increasing on the interval $(-\infty, 11)$ and decreasing on the interval $(11, \infty)$.

Therefore, the intervals of increase and decrease for the function are:
$\nearrow$ for $x < 11$; $\searrow$ for $x > 11$.

To determine the intervals where the function is increasing or decreasing, we first differentiate the function.

Given the function:

$y = \frac{1}{4}x^2 - \frac{7}{2}x$

Calculate the first derivative, $y'$, as follows:

$y' = \frac{d}{dx}\left(\frac{1}{4}x^2 - \frac{7}{2}x\right)$

Applying standard differentiation rules:

$y' = \frac{1}{4} \cdot 2x - \frac{7}{2}$

Simplifying this, we get:

$y' = \frac{1}{2}x - \frac{7}{2}$

Set the first derivative equal to zero to find the critical points:

$\frac{1}{2}x - \frac{7}{2} = 0$

Solving for $x$, we multiply the entire equation by 2 to clear the fractions:

$x - 7 = 0$

$x = 7$

This means that the function has a critical point at $x = 7$.

Evaluate the sign of $y'$ around the critical point to determine the intervals of increase and decrease:

• For $x < 7$, choose a test point like $x = 0$:
• $y'(0) = \frac{1}{2}(0) - \frac{7}{2} = -\frac{7}{2}$ (negative)
• For $x > 7$, choose a test point like $x = 8$:
• $y'(8) = \frac{1}{2}(8) - \frac{7}{2} = \frac{4 - 7}{2} = \frac{1}{2}$ (positive)

Therefore, the function is decreasing on the interval $(-\infty, 7)$ and increasing on the interval $(7, \infty)$.

From these analyses, we conclude:

The correct intervals are:

$\nearrow : x < 7$ (increasing)

$\searrow : x > 7$ (decreasing)

Thus, the correct answer choice is:

$\searrow : x > 7 ~~$.
$\nearrow : x < 7$

To determine where the given function increases or decreases, we follow these steps:

• Step 1: Differentiate the given function. Given the function $y = \frac{1}{2}x^2 + 4\frac{3}{5}x$, we first express $4\frac{3}{5}$ as an improper fraction, which is $\frac{23}{5}$. This makes the function $y = \frac{1}{2}x^2 + \frac{23}{5}x$.
• Step 2: Calculate the derivative of the function. The derivative $y'$ is computed as follows: $y' = \frac{d}{dx}\left(\frac{1}{2}x^2 + \frac{23}{5}x\right) = x + \frac{23}{5}.$
• Step 3: Set the derivative to zero to find critical points. Solve the equation $x + \frac{23}{5} = 0$: $x = -\frac{23}{5}.$
• Step 4: Identify intervals around the critical point. The critical point is $x = -\frac{23}{5}$, corresponding to a point of potential change from increasing to decreasing or vice versa.
• Step 5: Test the intervals determined by the critical points to find the sign of the derivative, determining whether the function is increasing or decreasing in those intervals.

- For $x < -\frac{23}{5}$, choose a test point such as $x = -5$:
$y'(-5) = -5 + \frac{23}{5} = -5 + 4.6 = -0.4$ which is negative, so the function is decreasing ($\searrow$).

- For $x > -\frac{23}{5}$, choose a test point such as $x = 0$:
$y'(0) = 0 + \frac{23}{5} = 4.6$ which is positive, indicating the function is increasing ($\nearrow$).

Therefore, the function decreases for $x < -\frac{23}{5}$ and increases for $x > -\frac{23}{5}$.

Thus, the intervals of increase and decrease for the function are:

$\searrow:x > -4\frac{3}{5}~~\\ \nearrow:x < -4\frac{3}{5}$

To find the intervals of increase and decrease for the function $y = \frac{1}{5}x^2 + 1\frac{1}{3}x$, we first calculate the derivative to analyze the behavior of the function.

Step 1: Finding the Derivative
The function is $y = \frac{1}{5}x^2 + \frac{4}{3}x$. To find the derivative, we use the power rule:

$y' = \frac{d}{dx}\left(\frac{1}{5}x^2 + \frac{4}{3}x\right) = \frac{2}{5}x + \frac{4}{3}$.

Step 2: Set the Derivative to Zero
To find critical points, set the derivative equal to zero:

$\frac{2}{5}x + \frac{4}{3} = 0$.

Solving for $x$, we multiply the equation by 15 (to eliminate fractions):

$6x + 20 = 0$.
$6x = -20$.
$x = -\frac{20}{6} = -\frac{10}{3}$.

So, the critical point is at $x = -\frac{10}{3}$, or $x = -3\frac{1}{3}$.

Step 3: Determine the Sign of the Derivative
Test the sign of the derivative on intervals around the critical point $x = -3\frac{1}{3}$:

• For $x < -3\frac{1}{3}$, choose $x = -4$. Then $y' = \frac{2}{5}(-4) + \frac{4}{3} = -\frac{8}{5} + \frac{4}{3}$. Compute to check if negative.
• For $x > -3\frac{1}{3}$, choose $x = 0$. Then $y' = \frac{2}{5}(0) + \frac{4}{3} = \frac{4}{3}$, which is positive.

Conclusion:
The function is decreasing ($\searrow$) for $x > -3\frac{1}{3}$ and increasing ($\nearrow$) for $x < -3\frac{1}{3}$.

Therefore, the correct intervals are:

$\searrow~:x > -3\frac{1}{3}$
$\nearrow~:x < -3\frac{1}{3}$

This matches with choice 4 of the provided options.

To find the intervals where the function $y = -4x^2 + 18x$ is increasing or decreasing, we need to first find its derivative.

The derivative of the function $y$ with respect to $x$ is:

$y' = \frac{d}{dx}(-4x^2 + 18x) = -8x + 18$.

Next, set the derivative to zero to find the critical points:

$-8x + 18 = 0$.

Solving this equation for $x$, we get:

$-8x = -18$.

$x = \frac{18}{8} = \frac{9}{4} = 2.25$.

This means $x = 2.25$ is a critical point, which corresponds to the vertex of the parabola.

Now, we need to determine the sign of $y'$ on either side of $x = 2.25$ to establish the intervals of increase and decrease.

• For $x < 2.25$, choose a test point such as $x = 0$:
  $y'(0) = -8(0) + 18 = 18$ (positive), indicating the function is increasing.
• For $x > 2.25$, choose a test point such as $x = 3$:
  $y'(3) = -8(3) + 18 = -24 + 18 = -6$ (negative), indicating the function is decreasing.

Therefore, the function is:

Increasing when $x < 2.25$.

Decreasing when $x > 2.25$.

Thus, the solution to the given problem is:

$\nearrow : x < 2\frac{1}{4}$ and $\searrow : x > 2\frac{1}{4}$.

To find the intervals where the function $y = -4x^2 + 28x$ increases or decreases, we will proceed with the following steps:

• Step 1: Differentiate the function to find $y'$.
• Step 2: Set $y'$ to zero and solve for $x$ to find critical points.
• Step 3: Use the critical points to define intervals on the x-axis.
• Step 4: Determine the sign of $y'$ in each interval to establish where the function increases or decreases.

Let's execute these steps in detail:

Step 1: Differentiate the function:
The function is given by $y = -4x^2 + 28x$. The derivative is calculated as follows:

$y' = \frac{d}{dx}(-4x^2 + 28x) = -8x + 28$.

Step 2: Find critical points where $y' = 0$:
Solve $-8x + 28 = 0$:

$-8x = -28$
$x = \frac{-28}{-8} = \frac{28}{8} = \frac{7}{2} = 3\frac{1}{2}$.

Step 3: Define intervals using the critical point $x = 3\frac{1}{2}$:
The intervals are $(-\infty, 3\frac{1}{2})$ and $(3\frac{1}{2}, \infty)$.

Step 4: Test the sign of $y'$ in each interval:

• For $x \in (-\infty, 3\frac{1}{2})$, choose a test point, say $x = 0$:
  $y'(0) = -8(0) + 28 = 28$. The derivative is positive, and the function is increasing.
• For $x \in (3\frac{1}{2}, \infty)$, choose a test point, say $x = 4$:
  $y'(4) = -8(4) + 28 = -32 + 28 = -4$. The derivative is negative, and the function is decreasing.

Therefore, the function is increasing on $(-\infty, 3\frac{1}{2})$ and decreasing on $(3\frac{1}{2}, \infty)$.

In conclusion, the intervals of increase and decrease are expressed as follows:

$\searrow~:x>3\frac{1}{2}~~\\\nearrow~:x<3\frac{1}{2}$

To find the intervals where the function $y = \frac{1}{9}x^2 + \frac{5}{3}x$ increases or decreases, we first compute its derivative.

Step 1: Differentiate the function with respect to $x$.

The derivative is: $y' = \frac{d}{dx} \left(\frac{1}{9}x^2 + \frac{5}{3}x\right) = \frac{2}{9}x + \frac{5}{3}$.

Step 2: Find critical points by setting $y' = 0$.

$\frac{2}{9}x + \frac{5}{3} = 0$.

Multiplying through by 9 to clear fractions: $2x + 15 = 0$.

Solve for $x$: $x = -\frac{15}{2} = -7.5$.

Step 3: Determine the sign of $y'$ on the intervals determined by the critical point $x = -7.5$.

Test values from each of the intervals $(-\infty, -7.5)$ and $(-7.5, \infty)$.

For $x < -7.5$: Choose $x = -8$. Compute $y'(-8)$:

$y'(-8) = \frac{2}{9}(-8) + \frac{5}{3} = -\frac{16}{9} + \frac{5}{3} = -\frac{16}{9} + \frac{15}{9} = -\frac{1}{9}$; which is negative.

For $x > -7.5$: Choose $x = -7$. Compute $y'(-7)$:

$y'(-7) = \frac{2}{9}(-7) + \frac{5}{3} = -\frac{14}{9} + \frac{15}{9} = \frac{1}{9}$; which is positive.

Therefore, the function decreases on the interval $x > -7.5$ and increases on the interval $x < -7.5$.

The correct interpretation in terms of the choices is:

$\searrow~:x > -7\frac{1}{2}$
$\nearrow~:x < -7\frac{1}{2}$