Word Count : 617

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To find the local minimum and local maximum values of the function \( f(x) = 3 - 5x^3 + 5x^4 - x^5 \), we follow these steps: ### Step 1: Find the derivative of \( f(x) \) The first derivative of \( f(x) \) will help us identify the critical points. \[ f'(x) = \frac{d}{dx} \left( 3 - 5x^3 + 5x^4 - x^5 \right) \] \[ f'(x) = -15x^2 + 20x^3 - 5x^4 \] ### Step 2: Find critical points by setting \( f'(x) = 0 \) We now solve \( f'(x) = 0 \) for the critical points. \[ -15x^2 + 20x^3 - 5x^4 = 0 \] \[ x^2 (-15 + 20x - 5x^2) = 0 \] This gives two possibilities: 1. _________ __________ _______ _______ ____ ________.
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