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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.3.73
Chapter 4, Problem 4.3.73

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).

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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.
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.
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.
Set the derivative equal to zero at x = 1: 2a(1) + b = 0, which simplifies to 2a + b = 0.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Finding Extrema Graphically Example 4

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.
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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.
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Substitution With an Extra Variable
Related Practice
Textbook Question

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.


y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)

Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dv/dt = (1/2)sec t tan t, v(0) = 1

Textbook Question

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = x³ / (3x² + 1)

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Textbook Question

30. Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.

Textbook Question

Theory and Examples

67. An inequality for positive integers Show that if a, b, c, and d are positive integers, then

[(a^2+1)(b^2+1)(c^2+1)(d^2+1)]/abcd ≥ 16

Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

80. y' = 1 - cot²θ, for 0 < θ < π