Math CentralQuandaries & Queries


Question from Russell, a student:

let f(x) = x^3 - 3a^2^ x +2a^4 with a parameter a > 1.

Find the coordinates of local minimum and local maximum

Find the coordinates of the inflection points

Hi Russell,

I am going to answer the same question but with the function f(x) = a2 x3 -6 a x + 5 and the restriction that a > 0.

To find the local maximum and minimum I need the derivative.

f '(x) = 3 a2 x2 - 6a = 3a(a x2 - 2)

Thus f '(x) = 0 if x = ±√(2/a). To determine if these give local maxima or minima I am going to use the second derivative test.

f "(x) = 6 a2x

When x >0, f "(x) >0 and when x < 0, f "(x) < 0. Thus there is a local minimum at x = √(2/a) and a local maximum at x = -√(2/a). Also since f "(x) changes sign at x = 0 there is an inflection point at x = 0.

Now try your function,

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