IB Maths AA SL 5.8 Extrema Optimization and Inflexion Questions

Practise IB Mathematics SL 5.8 by applying extrema, optimization and inflexion methods to questions from the Question Bank.

Exam points

Find local maxima and minima using first-derivative sign changes or second-derivative tests, and identify inflexion points and concavity from derivative information.
Form and solve optimization models for profit, area or volume, stating the feasible domain and interpreting the stationary solution in context.

[Maximum number: 7]

The derivative of a function f is given by $f^{\prime}(x)=\frac{2 x+2}{x^{2}+2 x+2}$ , for $x \in \mathbb{R}$ .

Hence, justify that the value of x found in part (b)(i) corresponds to a local minimum point on the graph of f.

It is given that $f(2)=3+\ln 10$ .

substituting x=-1 into $f^{\prime \prime}(x)$ , clearly leading to positive numerator A1

\begin{aligned} & f^{\prime \prime}(-1)=\left(\frac{-2+4}{1}\right)(=2) \\ & f^{\prime \prime}(-1)>0 \end{aligned}

therefore this is a local minimum point AG

Note: Do not award AOR1.

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