Sketch the graph of a cubic function that has three distinct real roots and a positive coefficient of x3x^3x3.
The vertical displacement, sss, of a mechanical component is modelled by the function s(t)=k+15at2−2t3s(t) = k + 15at^2 - 2t^3s(t)=k+15at2−2t3 where t≥0t \ge 0t≥0 is time, and aaa and kkk are constants with a>0a > 0a>0.
Show that the curve s(t)s(t)s(t) has a stationary point at its sss-intercept.
Given that the equation s(t)=0s(t) = 0s(t)=0 has three distinct real roots, determine the range of possible values for kkk in terms of aaa. You must use the second derivative to justify the nature of the stationary points used in your calculation.
Practise Edexcel A Level Maths 9.9 Using Second Derivatives with exam-style questions for A Level Maths. 57 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.