Questions: Using your graphing calculator, determine the interval for which the following function positive and determine the interval for which this function is increasing. f(x) = 2x^3 - 6x^2 - 26x + 30 Set the window to the following values: x Min = -10, x Max = 10, y Min = -100, y Max = 100 The interval for which this function is positive is: . U The interval for which this function is increasing is: U To enter infinity or negative infinity type inf or -inf in the blank. For decimal answers round to the nearest tenth (Eg: 1.7)

Using your graphing calculator, determine the interval for which the following function positive and determine the interval for which this function is increasing.

f(x) = 2x^3 - 6x^2 - 26x + 30

Set the window to the following values:

x Min = -10, x Max = 10, y Min = -100, y Max = 100

The interval for which this function is positive is: . U

The interval for which this function is increasing is:
U

To enter infinity or negative infinity type inf or -inf in the blank.

For decimal answers round to the nearest tenth (Eg: 1.7)

Transcript text: Using your graphing calculator, determine the interval for which the following function positive and determine the interval for which this function is increasing. \[ f(x)=2 x^{3}-6 x^{2}-26 x+30 \] Set the window to the following values: \[ x \operatorname{Min}=-10 \quad x \operatorname{Max}=10 \quad y \operatorname{Min}=-100 \quad y \operatorname{Max}=100 \] The interval for which this function is positive is: $\square$ . $\square$ U $\square$ $\square$ ) The interval for which this function is increasing is: $\square$ $\square$ U $\square$ $\square$ To enter infinity or negative infinity type inf or -inf in the blank. For decimal answers round to the nearest tenth (Eg: 1.7)

Solution

Solution Steps

Step 1: Find the Roots of the Function

To determine where the function $ f(x) = 2x^3 - 6x^2 - 26x + 30 $ is positive, we first find its roots by solving the equation $ f(x) = 0 $. The roots are $ x = -3 $, $ x = 1 $, and $ x = 5 $.

Step 2: Find the Derivative of the Function

Next, we compute the derivative of the function, given by $ f'(x) = 6x^2 - 12x - 26 $. This derivative will help us identify the critical points where the function may change from increasing to decreasing or vice versa.

Step 3: Find the Critical Points

We set the derivative equal to zero to find the critical points: \[ 6x^2 - 12x - 26 = 0 \] Solving this equation yields the critical points $ x = -1.3094010767585031 $ and $ x = 3.309401076758503 $.

Step 4: Determine Intervals of Positivity

To find the intervals where the function is positive, we evaluate the sign of $ f(x) $ in the intervals defined by the roots: