Given y= x 4 4x 3 + 2. Find the critical numbers, maxima, minima, and inflection

Question

Given y= x4 4x3 + 2.  Find the critical numbers, maxima, minima, and inflection points of this function.

Explanation / Answer

y = x^4 - 4x^3 + 2 dy/dx = 4x^3 - 12x^2 d^2y/dx^2 = 12x^2 - 24x putting, dy/dx = 0 4x^2(x - 3) = 0 => x = 0 or x = 3 putting d^2y/dx^2 = 0 12x^2 - 24x = 0 12x(x - 2) = 0 x = 0 or x = 2 crictical numbers are, x = 0, 2 and 3 at x = 0, d^2y/dx^2 = 0 it is a point of inflection at x = 2, d^2y/dx^2 = 0 it is also a point of inflection at x = 3 d^2y/dx^2 = 12*3^2 - 24*3 = 36 (>0) hence it is th minimum point hence minima at x = 3 and points of inflection at x = 0 and 2

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