Question
Determine the local minimum and local maximum values of the function f defined by .
Answer :
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To find the local minima and maxima of ( f(x) = 3 - 5x^3 + 5x^4 - x^5 ), we first calculate the derivative: [ f'(x) = \frac{d}{dx}\big(3 - 5x^3 + 5x^4 - x^5\big) = -15x^2 + 20x^3 - 5x^4 ] Factor out (-5x^2): [ f'(x) = -5x^2(3 - 4x + x^2) ] Set ( f'(x) = 0 ) to find critical points: [ -5x^2(3 - 4x + x^2) = 0 ] This gives: [ x^2 = 0 \implies x = 0 ] [ x^2 - 4x + 3 = 0 \implies x = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2} ] [ x = 3 \quad
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