Determine the value(s) of x where the function will have an instantaneous rate of change of 14. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x) = x3 + 33 x2 + 14 x + 39

f'= 3x^2+66x+14=14

3x^2+66x=0
x=0, or x=-22

check that.

To find the value(s) of x where the function has an instantaneous rate of change of 14, we need to find the derivative of the function and set it equal to 14. Let's go step-by-step:

Step 1: Find the derivative of the function f(x) = x^3 + 33x^2 + 14x + 39.

The derivative of f(x) is given by f'(x) = 3x^2 + 66x + 14.

Step 2: Set the derivative f'(x) equal to 14.

3x^2 + 66x + 14 = 14.

Step 3: Simplify the equation.

3x^2 + 66x = 0.

Step 4: Factor out the common factor of x.

x(3x + 66) = 0.

Step 5: Set each factor equal to zero and solve for x.

x = 0 or 3x + 66 = 0.

From the second equation, we can solve for x:

3x + 66 = 0.

Subtract 66 from both sides:

3x = -66.

Divide both sides by 3:

x = -22.

So, the solution to the equation is x = -22 or x = 0.

Therefore, the value(s) of x where the function has an instantaneous rate of change of 14 are -22 and 0.

To find the value(s) of x where the function has an instantaneous rate of change of 14, we need to find the derivative of the function and set it equal to 14. Let's go through the steps:

1. Take the derivative of the function f(x) with respect to x.
f'(x) = 3x^2 + 66x + 14

2. Set f'(x) equal to 14 and solve for x:
3x^2 + 66x + 14 = 14

3. Simplify the equation and move all terms to one side:
3x^2 + 66x = 0

4. Factor out the common factor of x on the left side:
x(3x + 66) = 0

5. Set each factor equal to zero and solve for x:
x = 0 or 3x + 66 = 0

For the first factor x = 0, there is a solution: x = 0.

For the second factor 3x + 66 = 0, solve for x:
3x = -66
x = -22

So the values of x where the function has an instantaneous rate of change of 14 are x = 0 and x = -22.