



 
Hi Jyotishman, This problem involves geometry and calculus. The geometry tells me what it is I need to find and the calculus gives me a method to find it. I can get you started. First I graphed y = f(x) = x^{3}  13x + a with a = 0. The axes are in red and I didn't put the scale on the axes since I am interested in the shape not the scale. This graph crosses the xaxis in 3 places marked by the red dots. As a increases he graph moves upwards relative to the axes maintaining three distinct zeros until until a attains a value for which the polynomial f(x) = x^{3}  13x + a two distinct roots, one of them I am going to call p as in the diagram. (One root is not on my diagram.) O If a increases beyond this value the resulting graph only crosses the xaxis at one point (not on the diagram). The task is thus to find p and then the value of a so that f(p) = p^{3}  13p + a = 0. p is a critical value for the polynomial f(x) so it can be found by solving f '(x) = 0. Do you now see how to complete the problem? If you need further assistance write back. Harley  


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