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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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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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Related Practice
Textbook Question

Finding Position from Velocity or Acceleration


Exercises 45–48 give the acceleration a=d²s/dt², initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t.


a = 9.8, v(0) = −3, s(0) = 0

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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.

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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

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Textbook Question

Applications


Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?

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Textbook Question

Finding Functions from Derivatives


Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.

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Textbook Question

32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.

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