Question
Determine the local minimum and local maximum values of the function f defined by f(x) = 3 - 5x3 + 5x4 - x5.
Answer :
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Skip in 10
We are asked to find the local minima and maxima of the function: $$ f(x) = 3 - 5x^3 + 5x^4 - x^5 $$ We will solve this manually, step by step. --- ### Step 1: Compute the derivative Local extrema occur where $f'(x) = 0$. $$ f(x) = 3 - 5x^3 + 5x^4 - x^5 $$ $$ f'(x) = -15x^2 + 20x^3 - 5x^4 $$ Factor $-5x^2$: $$ f'(x) = -5x^2 (3 - 4x + x^2) $$ --- ### Step 2: Solve $f'(x) = 0$ $$ -5x^2 (x^2 - 4x + 3) = 0 $$ 1. $-5x^2 = 0$ → $x = 0$ 2. $x^2 - 4x + 3 = 0$ → Solve quadratic: $$ x^2 - 4x + 3 = 0 $$ $$ x = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm \sqrt{4}}{2} = \frac{4 \pm 2}{2} $$ $$ x = 1 \quad \text{or} \quad x = 3 $$
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