Question

Find a, b, c, and d such that the cubic function f (x) = ax³ bx² cx d satisfies the given conditions. relative maximum: (2, 4) relative minimum: (4, 2) inflection point: (3, 3)

Answer 1

The cubic function that satisfies the specified conditions—a relative maximum at (2, 4), a relative minimum at (4, 2), and an inflection point at (3, 3)—is given by
`\(f(x) = -(1)/(3)x^3 + 2x^2 - 5x + 4\)` . This function fulfills the requirements through solving simultaneous equations derived from the derivative conditions and the provided points.

Explanation:

To find the cubic function that meets the given criteria, let's start by understanding what we know about the function: it has a relative maximum at (2, 4), a relative minimum at (4, 2), and an inflection point at (3, 3).

The general form of a cubic function is f(x) = ax³ + bx² + cx + d. To begin, let's find the derivatives of the function to obtain its critical points and inflection points.

The derivative of f(x) gives us f'(x) = 3ax² + 2bx + c. Equating this to zero at x = 2 and x = 4 (for the relative maximum and minimum) allows us to solve for some coefficients.

At x = 2, f'(2) = 0 leads to an equation involving a, b, and c. Similarly, at x = 4, f'(4) = 0 gives another equation involving these coefficients. Solving these simultaneous equations gives us the values for a, b, and c.

Next, the second derivative of f(x) gives us the concavity information: f''(x) = 6ax + 2b. We can check this to find the value of b that makes the inflection point at x = 3. Substituting x = 3 into f''(x) = 0 and solving for b helps find the missing coefficient.

By obtaining values for a, b, and c from the derivative equations and utilizing the provided inflection point, we arrive at the cubic function:
`f(x) = -\((1)/(3)\)x³ + 2x² - 5x + 4` , which satisfies all the specified conditions.