A curve represents the thermal expansion of a composite alloy, where the length LLL is defined as a function of the temperature θ\thetaθ such that L=f(θ)L = f(\theta)L=f(θ). It is observed that the graph has a point of inflection at θ=185\theta = 185θ=185. Given that f′(185)=αf'(185) = \alphaf′(185)=α and f′′(185)=βf''(185) = \betaf′′(185)=β, where α\alphaα and β\betaβ are constants, identify which of the following statements must be true:
f′(185)=0 f'(185) = 0 f′(185)=0 f′′(185)=0 f''(185) = 0 f′′(185)=0 f′(185)≠0 f'(185) \neq 0 f′(185)=0 f′′(185)<0 f''(185) < 0 f′′(185)<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.