The power consumption PPP (in kW) of an industrial extractor fan is modelled by the equation
P=13w3−6w+kw,w>0 P = \frac{1}{3}w^3 - 6w + \frac{k}{w}, \quad w > 0 P=31w3−6w+wk,w>0where www is the angular velocity in rad/s and kkk is a constant. The fan is observed to have a stationary point in its power profile when w=2w = 2w=2.
Show that k=−8k = -8k=−8.
Determine the nature of the stationary point when w=2w = 2w=2, justifying your answer.
The power profile has a second stationary point.
Using algebra, find the exact value of the angular velocity www at this second stationary point.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.