



 
Hi Russell, I am going to answer the same question but with the function f(x) = a^{2} x^{3} 6 a x + 5 and the restriction that a > 0. To find the local maximum and minimum I need the derivative.
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.
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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