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.
375 exam-style questions on AQA A Level Maths 1.10 G: Differentiation, 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). Each one has a worked solution and a mark scheme showing where the marks go.