Asked by BlUbArRy!2#
I chose cubic function f(x)=x^3
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Answered by
GPT-4o mini
Great choice! The cubic function \( f(x) = x^3 \) has some interesting properties. Here are a few key points about this function:
1. **Shape and Graph**: The graph of \( f(x) = x^3 \) is a smooth curve that passes through the origin (0,0) and has symmetry about the origin. It increases without bound as \( x \) increases and decreases without bound as \( x \) decreases.
2. **Intercepts**: The function has one intercept, which is at the origin \( (0, 0) \).
3. **Increasing and Decreasing**: The function is always increasing; it has no local maxima or minima. This is because the derivative \( f'(x) = 3x^2 \) is non-negative for all real \( x \) and is zero only at \( x = 0 \).
4. **Inflection Point**: The function has an inflection point at \( x = 0 \). This is where the concavity changes. The second derivative \( f''(x) = 6x \) is zero at this point.
5. **End Behavior**: As \( x \) approaches \( \infty \), \( f(x) \) approaches \( \infty \), and as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \).
Would you like to explore any specific properties or concepts related to this cubic function?
1. **Shape and Graph**: The graph of \( f(x) = x^3 \) is a smooth curve that passes through the origin (0,0) and has symmetry about the origin. It increases without bound as \( x \) increases and decreases without bound as \( x \) decreases.
2. **Intercepts**: The function has one intercept, which is at the origin \( (0, 0) \).
3. **Increasing and Decreasing**: The function is always increasing; it has no local maxima or minima. This is because the derivative \( f'(x) = 3x^2 \) is non-negative for all real \( x \) and is zero only at \( x = 0 \).
4. **Inflection Point**: The function has an inflection point at \( x = 0 \). This is where the concavity changes. The second derivative \( f''(x) = 6x \) is zero at this point.
5. **End Behavior**: As \( x \) approaches \( \infty \), \( f(x) \) approaches \( \infty \), and as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \).
Would you like to explore any specific properties or concepts related to this cubic function?
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