Find a cubic function, in the form below, that has a local maximum value of 3 at -2 and a local minimum value of 0 at 1.
f (x) = ax3 + bx2 + cx + d
*Fixed*
Find a cubic function, in the form below, that has a local maximum value of 3 at -2 and a local minimum value of 0 at 1.
f (x) = ax^3 + bx^2 + cx + d
D=7. C=2. B =5/7 .A=-2/9
august ames
To find a cubic function in the given form with the specified local maximum and minimum values, we can start by using the information about the local maximum and minimum to create two equations.
1. Local Maximum: To find the local maximum, the derivative of the cubic function should be equal to zero. Since the cubic function is f(x) = ax^3 + bx^2 + cx + d, the derivative would be f'(x) = 3ax^2 + 2bx + c. At the x-coordinate of the local maximum, which is -2, the derivative should be zero. Therefore, we have the equation:
3a(-2)^2 + 2b(-2) + c = 0
12a - 4b + c = 0 --------(Equation 1)
2. Local Minimum: Similar to the local maximum, at the x-coordinate of the local minimum, which is 1, the derivative should be zero. So, we have another equation:
3a(1)^2 + 2b(1) + c = 0
3a + 2b + c = 0 --------(Equation 2)
Now, we have two equations (Equation 1 and Equation 2) with three unknowns (a, b, c), which means we need one more piece of information to solve the system and find the values of a, b, and c.
To find this additional piece of information, we can consider either the value of the cubic function or its derivative at a specific point. Let's consider the value of the cubic function, f(x), at a convenient point.
We know that the local maximum value of the cubic function is 3 at x = -2. Given the cubic function, we can set up the equation:
f(-2) = a(-2)^3 + b(-2)^2 + c(-2) + d = 3
By substituting the values, we obtain:
-8a + 4b - 2c + d = 3 --------(Equation 3)
Now we have a system of three equations (Equation 1, Equation 2, and Equation 3) with three unknowns (a, b, c). We can solve this system of equations to find the values for a, b, c, and d.
After solving the system of equations, you will have the values of a, b, c, and d, which can be used to write the cubic function in the desired form, f(x) = ax^3 + bx^2 + cx + d.