f(x) = x^3 + 2
Understand the Problem
The question is likely asking how to work with the function given by f(x) = x³ + 2, which may involve finding its properties, such as its derivative, evaluating it at a certain point, or analyzing its behavior.
Answer
The function's derivative is \( f'(x) = 3x^2 \), and it evaluates to \( f(1) = 3 \) with a critical point at \( x = 0 \) where \( f(0) = 2 \).
Answer for screen readers
The function ( f(x) = x^3 + 2 ) has a derivative ( f'(x) = 3x^2 ), which is always non-negative. At ( x = 1 ), ( f(1) = 3 ), and the critical point is at ( x = 0 ) with ( f(0) = 2 ).
Steps to Solve
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Identify the function We are given the function ( f(x) = x^3 + 2 ).
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Find the derivative To analyze the behavior of ( f(x) ), we first find the derivative ( f'(x) ). The derivative of ( x^3 ) is ( 3x^2 ).
Thus, $$ f'(x) = 3x^2 $$
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Evaluate the function at a specific point To evaluate ( f(x) ) at a certain point, substitute the point into the function. For example, let's evaluate it at ( x = 1 ): $$ f(1) = 1^3 + 2 = 1 + 2 = 3 $$
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Analyze the behavior The derivative ( f'(x) = 3x^2 ) is always non-negative. This means that the function is always increasing since ( f'(x) > 0 ) for all ( x \neq 0 ).
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Find critical points The only critical point occurs when ( f'(x) = 0 ): $$ 3x^2 = 0 $$ Thus, ( x = 0 ) is a critical point.
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Evaluate the function at the critical point Now, substituting ( x = 0 ): $$ f(0) = 0^3 + 2 = 2 $$
The function ( f(x) = x^3 + 2 ) has a derivative ( f'(x) = 3x^2 ), which is always non-negative. At ( x = 1 ), ( f(1) = 3 ), and the critical point is at ( x = 0 ) with ( f(0) = 2 ).
More Information
The function ( f(x) = x^3 + 2 ) is a cubic polynomial, which typically has one local minimum or maximum but in this case, does not due to its always-increasing nature. This illustrates how polynomial functions can behave differently based on their degrees.
Tips
- Confusing critical points: Remember that critical points occur where the derivative equals zero or is undefined. For polynomials, this typically happens where ( f'(x) = 0 ).
- Ignoring the behavior of the derivative: Always analyze the derivative to understand the function's increasing or decreasing nature.
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