The profit PPP, in thousands of pounds, generated by a chemical plant is modeled by the equation
P=54v12−2v32+10,v>0 P = 54v^{\frac{1}{2}} - 2v^{\frac{3}{2}} + 10, \quad v > 0 P=54v21−2v23+10,v>0where v v\,v is the volume of a specific catalyst used in litres.
Find dPdv\dfrac{dP}{dv}dvdP, giving each term in its simplest form.
Hence find the coordinates of the stationary point of the profit model.
Find d2Pdv2\dfrac{d^2 P}{dv^2}dv2d2P and hence determine the nature of this 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.