Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function f(x) = √x has a local maximum on the interval [0,∞).
5. Graphical Applications of Derivatives
Intro to Extrema
5:53 minutesProblem 4.1.89b
Textbook Question
Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.
b. Does either g or h have a local extreme value at x = 2? Explain.
Verified step by step guidance1
To determine if g(x) or h(x) has a local extreme value at x = 2, we need to find their derivatives and evaluate them at x = 2.
First, find the derivative of g(x) = x * f(x) + 1. Use the product rule for differentiation: if u(x) = x and v(x) = f(x), then the derivative g'(x) = u'(x)v(x) + u(x)v'(x).
Calculate g'(x): g'(x) = 1 * f(x) + x * f'(x) = f(x) + x * f'(x).
Evaluate g'(x) at x = 2: g'(2) = f(2) + 2 * f'(2). Since f(2) = 0 and f has a local extreme at x = 2, f'(2) = 0. Thus, g'(2) = 0.
Now, find the derivative of h(x) = x * f(x) + x + 1. Differentiate h(x) using the sum and product rules: h'(x) = f(x) + x * f'(x) + 1. Evaluate h'(x) at x = 2: h'(2) = f(2) + 2 * f'(2) + 1. Since f(2) = 0 and f'(2) = 0, h'(2) = 1, indicating h does not have a local extreme at x = 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Local Extreme Values
A local extreme value of a function occurs at a point where the function reaches a maximum or minimum relative to its immediate surroundings. For a function to have a local extreme value at a point, the derivative at that point must be zero, indicating a horizontal tangent. In this case, since f has a local extreme at x = 2, we know that f'(2) = 0.
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Differentiability and Derivatives
A function is differentiable at a point if it has a defined derivative there, which means it is smooth and has no sharp corners or discontinuities. The derivative provides information about the function's rate of change and is crucial for determining local extreme values. Since f is differentiable everywhere, we can analyze the derivatives of g and h to assess their behavior at x = 2.
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Product Rule in Differentiation
The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are functions, the derivative of their product is given by u'v + uv'. This rule is essential for differentiating g(x) = xf(x) and h(x) = xf(x) + x + 1, as both functions involve the product of x and f(x), allowing us to determine if they have local extreme values at x = 2.
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