Critical Points of a Quartic

Calculus Level 2

Classify the critical points of f ( x ) = x 4 − 4 x 3 + 16 x f(x) = x^4 - 4x^3 + 16x .

x = − 1 x = -1 is a local maximum, and x = 2 x = 2 is a local minimum. x = − 1 x = -1 is a local maximum, and x = 2 x = 2 is an inflection point. x = − 1 x = -1 is a local minimum, and x = 2 x = 2 is an inflection point. x = − 1 x = -1 is a local minimum, and x = 2 x = 2 is a local maximum. x = − 1 x = -1 is an inflection point, and x = 2 x = 2 is a local maximum.

Henry Maltby by Henry Maltby , 26, USA

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1 solution

Henry Maltby
May 5, 2016

The derivative of f f is f ′ ( x ) = 4 x 3 − 12 x 2 + 16 = 4 ( x + 1 ) ( x − 2 ) 2 , f'(x) = 4x^3 - 12x^2 + 16 = 4(x + 1)(x - 2)^2, so the derivative is zero at x = − 1 x = -1 and x = 2 x = 2 . Since f ′ f' is defined on all real numbers, the only critical points of the function are x = − 1 and x = 2. \boxed{x = -1 \text{ and } x = 2.}

The second derivative of f f is f ′ ′ ( x ) = 12 x 2 − 24 x , f''(x) = 12x^2 - 24x, so x = − 1 x = -1 is a local minimum by the Second Derivative Test. x = 2 x = 2 is an inflection point, because f ′ ′ f'' is negative beforehand and positive afterwards.

There is a local minimum at \(x = -1\) and a change in concavity at \(x = 2\). There is a local minimum at x = − 1 x = -1 and a change in concavity at x = 2 x = 2 .

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