The power consumption, www (in watts), of a submersible exploration drone at depth ddd (in meters) is modelled by the equation
w=3d3−40d+kd,d>0 w = 3d^3 - 40d + \frac{k}{d}, \quad d > 0 w=3d3−40d+dk,d>0where kkk is a constant. The drone has a stationary point in its power consumption at depth PPP where d=2d = 2d=2.
Show that k=−16k = -16k=−16.
Determine the nature of the stationary point at PPP, justifying your answer.
The power consumption model suggests there is a second stationary point at a different depth.
Using algebra, find the depth of this second stationary point.
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.