Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x³ + 4x² + 7, (−∞, 0)
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4. Applications of Derivatives
Differentials
2:49 minutesProblem 27
Textbook Question
Finding Functions from Derivatives
Suppose that f(−1) = 3 and that f'(x) = 0 for all x. Must f(x) = 3 for all x? Give reasons for your answer.
Verified step by step guidance1
First, understand the meaning of the derivative f'(x) = 0 for all x. This implies that the function f(x) has a constant slope of 0, meaning it is a horizontal line.
Since the derivative represents the rate of change of the function, a derivative of 0 indicates that the function does not change as x changes. Therefore, f(x) must be a constant function.
Given that f(x) is a constant function, it must take the same value for all x.
We are provided with the information that f(−1) = 3. Since f(x) is constant, this value must be the same for all x.
Thus, we conclude that f(x) = 3 for all x, because a constant function with f(−1) = 3 implies that the constant value is 3 everywhere.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Constant Function
A constant function is a function that always returns the same value, no matter the input. In mathematical terms, if f'(x) = 0 for all x, it implies that the function f(x) does not change as x changes, indicating that f(x) is a constant function.
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Derivative
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. If the derivative f'(x) is zero for all x, it means the function has no slope and is flat, suggesting that the function is constant across its domain.
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Initial Condition
An initial condition provides specific information about a function at a particular point, which helps determine the constant of integration when solving differential equations. In this problem, f(−1) = 3 serves as an initial condition, confirming that the constant value of the function f(x) is 3.
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