Sketch the graph of any cubic function that has three distinct real roots and a positive coefficient of x3x^3x3.
A research team models the potential energy VVV of a chemical system as a function of its configuration xxx using the equation
V(x)=2x3−9ax2+k V(x) = 2x^3 - 9ax^2 + k V(x)=2x3−9ax2+kwhere aaa and kkk are constants and a>0a > 0a>0.
Verify that the energy profile has a stationary point where it intersects the vertical VVV-axis.
Given that the equation V(x)=0V(x) = 0V(x)=0 has three distinct real roots, determine the range of possible values for kkk in terms of aaa by considering the nature and positions of the local maximum and local minimum points.
Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 327 questions covering 12.1 Gradients of Curves, 12.2 Differentiation from first principles, 12.3 Differentiating x^n, 12.4 Differentiating Quadratics, 12.5 Differentiating functions with two or more terms, 12.6 Gradients, Tangents and Normals, 12.7 Increasing and Decreasing Functions, 12.8 Second Order Derivatives, 12.9 Stationary Points, 12.10 Sketching Gradient Functions, and 12.11 Modelling with Differentiation, matched to the Edexcel A Level Maths (9MA0) 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.
