The concentration, CCC, of a reactant in a non-isothermal chemical process is modeled by the function C=ϕ(t)C = \phi(t)C=ϕ(t), where ttt is the time in seconds. It is known that the graph of CCC against ttt has a point of inflection at t=12t = 12t=12. Given that ϕ′(12)=a\phi'(12) = aϕ′(12)=a and ϕ′′(12)=b\phi''(12) = bϕ′′(12)=b, where aaa and bbb are constants,
identify which of the following statements must be true:
ϕ′(12)=0 \phi'(12) = 0 ϕ′(12)=0 ϕ′′(12)=0 \phi''(12) = 0 ϕ′′(12)=0 ϕ′(12)≠0 \phi'(12) \neq 0 ϕ′(12)=0 ϕ′′(12)>0 \phi''(12) > 0 ϕ′′(12)>0425 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.