Sketch the graph of any cubic function that has both three distinct real roots and a negative coefficient of x3x^3x3.
The function g(x)g(x)g(x) is defined by
g(x)=x3−4ax2+k g(x) = x^3 - 4ax^2 + k g(x)=x3−4ax2+kwhere aaa and kkk are constants and a>0a > 0a>0.
Show that there is a stationary point where the curve crosses the yyy-axis.
Given that the equation g(x)=0g(x) = 0g(x)=0 has three distinct real roots, find the range of possible values for kkk in terms of aaa by considering the positions of the local maximum and local minimum points.
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.
