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1 Pure Mathematics

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Question 174

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)<0
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