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.
717 exam-style questions on Edexcel A Level Maths Differentiation, covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change. Each one has a worked solution and a mark scheme showing where the marks go.