Question

Determine the values of constants a,b,c,and d so that f(x)=ax³+bx²+cx+d has a local minimum at the point (0,0) and a local maximum at the point (1,2).

Answer 1

The constants for the cubic function with a local minimum at (0,0) and local maximum at (1,2) are determined by solving a system of equations derived from the conditions of the function and its derivatives at those points. The constants are a = 2, b = 0, c = 0, and d = 0, resulting in the function f(x) = 2x³.

Step-by-step explanation:

To determine the values of the constants a, b, c, and d for the cubic function f(x) = ax³ + bx² + cx + d with a local minimum at the point (0,0) and a local maximum at the point (1,2), we need to use the given conditions and the properties of derivatives for local extrema.

Conditions for Extrema:

1. f(0) = d = 0 (function value at x=0).
2. f(1) = a + b + c + d = 2 (function value at x=1).
3. f'(x) = 3ax² + 2bx + c (the derivative of f(x)).
4. f'(0) = c = 0 (derivative at the local minimum x=0).
5. f'(1) = 3a + 2b + c = 0 (derivative at the local maximum x=1).

Applying these conditions, we get a system of equations:

• d = 0
• a + b + c + d = 2
• c = 0
• 3a + 2b + c = 0

From these, we can determine:

• d = 0
• c = 0
• And, substituting c and d into the equations, we have:
• a + b = 2
• 3a + 2b = 0

Solving the system of equations for a and b, we find:

• a = 2
• b = 0

Therefore, the cubic function is f(x) = 2x³.