Theory and Examples


Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).

Verified step by step guidance

1. First, understand that for the function f(x) = ax² + bx to have an absolute maximum at the point (1,2), the point (1,2) must lie on the curve of the function. This means f(1) = 2.
2. Substitute x = 1 into the function: f(1) = a(1)² + b(1) = a + b. Set this equal to 2, giving the equation: a + b = 2.
3. Next, for the point (1,2) to be an absolute maximum, the derivative of the function, f'(x), must be zero at x = 1. Calculate the derivative: f'(x) = 2ax + b.
4. Set the derivative equal to zero at x = 1: 2a(1) + b = 0, which simplifies to 2a + b = 0.
5. Now, solve the system of equations: a + b = 2 and 2a + b = 0. Use these equations to find the values of a and b that satisfy both conditions.

Key Concepts

Absolute Maximum

An absolute maximum of a function is the highest point over its entire domain. For a function f(x), it occurs at a point x = c if f(c) ≥ f(x) for all x in the domain. In this problem, the function f(x) = ax² + bx must have its highest value at the point (1,2), meaning f(1) = 2 and f(x) ≤ 2 for all x.

Derivative and Critical Points

The derivative of a function, f'(x), provides the slope of the tangent line at any point x. Critical points occur where f'(x) = 0 or is undefined, indicating potential maxima, minima, or inflection points. To find where f(x) = ax² + bx has an absolute maximum, we need to find its derivative, set it to zero, and solve for x to identify critical points.

Substitution and System of Equations

To determine the constants a and b, substitute the given point (1,2) into the function, ensuring f(1) = 2. This creates an equation involving a and b. Additionally, use the condition from the derivative that ensures a maximum at x = 1, leading to a system of equations. Solving this system will yield the values of a and b that satisfy both conditions.