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+15at2−2t3 s(t) = k + 15at^2 - 2t^3 s(t)=k+15at2−2t3where 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 polynomial equation s(t)=0s(t) = 0s(t)=0, when extended to all real values of ttt, 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.
57 exam-style questions on Edexcel A Level Maths 9.9 Using Second Derivatives. Each one has a worked solution and a mark scheme showing where the marks go.