Find \( f^{\prime}(x) \). \[ f(x)=4 x^{15}-8 x^{3}+8 \] \( f^{\prime}(x)= \) Compare the graphs of \( f \) and \( f^{\prime} \) and use them to explain why your answer is reasonable. \( f^{\prime}(x)=0 \) when \( f \) Select--- \( f^{\prime} \) is positive when \( f \) Select-- \( f^{\prime} \) is negative when \( f \) Select---
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Bonus Knowledge
To find \( f'(x) \), we differentiate \( f(x) = 4x^{15} - 8x^3 + 8 \). Applying the power rule, we have: \[ f'(x) = 60x^{14} - 24x^2 \] Now, let's discuss the relationship between the functions \( f \) and \( f' \). The graph of \( f \) will show where it increases and decreases based on the sign of \( f' \). When \( f'(x) = 0 \), this indicates the critical points where the function \( f \) has either a local maximum or minimum—essentially where the slope of \( f \) flattens out. Meanwhile, if \( f' > 0 \), it means that \( f \) is increasing. Conversely, when \( f' < 0 \), \( f \) is decreasing. So, identifying these intervals allows us to characterize the behavior of the original function effectively. To summarize: - \( f'(x) = 0 \) when \( f \) is at critical points. - \( f' > 0 \) when \( f \) is increasing. - \( f' < 0 \) when \( f \) is decreasing.
