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 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.
