Question

Find a, b, c, and d such that the cubic function

f(x) = ax³ + bx² + cx + d

satisfies the given conditions.

Relative maximum: (3, 23)
Relative minimum: (5, 21)
Inflection point: (4, 22)

Answer 1

To find the values of a, b, c, and d in the cubic function, substitute the given points into the cubic function and solve the resulting system of equations.

Step-by-step explanation:

To find the values of a, b, c, and d in the cubic function f(x) = ax³ + bx² + cx + d, we can use the given information about the relative maximum, relative minimum, and inflection point.

Step 1: Using the fact that the relative maximum occurs at (3, 23), we can substitute x = 3 and f(x) = 23 into the cubic function to get the equation 27a + 9b + 3c + d = 23.

Step 2: Similarly, using the other two given points, we can obtain two more equations. Substituting x = 5 and f(x) = 21, we get 125a + 25b + 5c + d = 21. And substituting x = 4 and f(x) = 22, we get 64a + 16b + 4c + d = 22.

Step 3: Solving the system of equations formed by the three equations obtained in steps 1 and 2 will give us the values of a, b, c, and d.