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 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 AQA A Level Maths 1.10 G: Differentiation with exam-style questions for A Level Maths. 321 questions covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only), matched to the AQA A Level Maths (7357) 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.
