IB Maths AI HL Sl 5 6 Stationary Points Questions

Practise solving f′(x) = 0 for stationary points, classifying local extrema and distinguishing them from absolute extrema on a stated domain.

Exam points

Solve f′(x)=0 to find stationary-point x-values, using technology where appropriate.
Classify local maxima and minima from derivative behaviour or the graph.
Distinguish local from absolute extrema on a stated domain and interpret the result.

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Study SL 5.6—Stationary points conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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Linda owns a field, represented by the shaded region R. The plan view of the field is shown in the following diagram, where both axes represent distance and are measured in metres.

Figure for Question IB Maths AI HL Sl 5 6 Stationary Points Questions question 1 — IB Maths AI HL

The segments [AB], [CD] and [AD] respectively represent the western, eastern and southern boundaries of the field. The function, f(x), models the northern boundary of the field between points B and C and is given by

f(x)=\frac{-x^{2}}{50}+2 x+30, \text { for } 0 \leq x \leq 70

Hence find the coordinates of the point on the field that is furthest north.

Point A has coordinates (0,0), point B has coordinates (0,30), point C has coordinates (70,72) and point D has coordinates (70,0).

$0=\frac{-x}{25}+2$ OR sketch of $f^{\prime}(x)$ with x-intercept indicated M1
$x=50 \quad$ A1
$y=80 \quad$ A1
(50,80)
Note: Award M0A0A1 for the coordinate (50,80) seen either with no working or found from a graph of f(x).

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IB Maths AI HL Sl 5 6 Stationary Points Questions question 1