The function you provided is \( f(x) = -2x^3 - 1 \). This is a cubic function due to the \( x^3 \) term. Here are some properties and characteristics of this function:
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Graph Shape: The graph of \( f(x) \) will have the general shape of a cubic function. Since the leading coefficient (-2) is negative, the ends of the graph will point downwards.
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Intercepts:
- Y-intercept: To find the y-intercept, set \( x = 0 \): \[ f(0) = -2(0)^3 - 1 = -1 \] So, the y-intercept is \( (0, -1) \).
- X-intercepts: To find the x-intercepts, set \( f(x) = 0 \): \[ -2x^3 - 1 = 0 \quad \Rightarrow \quad -2x^3 = 1 \quad \Rightarrow \quad x^3 = -\frac{1}{2} \] Thus, \( x = -\sqrt[3]{\frac{1}{2}} \).
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End Behavior:
- As \( x \to \infty \), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).
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Derivative:
- To find the critical points, you can take the derivative \( f'(x) \): \[ f'(x) = -6x^2 \] This derivative will be zero when \( x = 0 \). Since \( f'(x) \leq 0 \) for all \( x \), the function is decreasing everywhere.
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Behavior:
- The function has no local maxima or minima, since it is always decreasing.
If you have any specific questions or need further details about this function (like plotting it, finding limits, etc.), feel free to ask!