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Longhand

Before the final

Reporting a critical point without classifying it

The habit: A zero derivative marks a candidate: a sign change of f′ says which kind, and so does f″ when it is not 0; with no sign change it is neither.

5 practice questions are built to catch it, in 3 topics.

See it fail in a figure: Flat is not a turn, in the curve sketching lesson

Practise against it: 4 questions

What it looks like

A question from optimization, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

Find the coordinates of the local minimum of h(x)=x3−27x+5h(x) = x^{3} - 27x + 5.

With the slip
(−3,59)\displaystyle (-3, 59)
Worked through
(3,−49)\displaystyle (3, -49)

h′(−3) = 0, but h″(−3) = −18 is negative, so (−3, 59) is the local maximum. Test the other root of h′ the same way.

See the worked solution

Where it comes up

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

  1. 01
    Find the absolute maximum value of f(x)=x3−6x2+9x+2f(x) = x^{3} - 6x^{2} + 9x + 2 on the closed interval [0,5][0, 5].

    OptimizationStandard

  2. 02
    How many of the critical points of f(x,y)=x3−3x+y3−3y2f(x, y) = x^{3} - 3x + y^{3} - 3y^{2} are saddle points?

    Optimization in two variables and Lagrange multipliersStandard

  3. 03
    A thin wire is bent into the circle x2+y2=20x^{2} + y^{2} = 20, with lengths in centimetres. The temperature at the point (x,y)(x, y) of the wire is T(x,y)=30+2x−yT(x, y) = 30 + 2x - y degrees Celsius. Use Lagrange multipliers to find the lowest temperature on the wire.

    Optimization in two variables and Lagrange multipliersStandard

  4. 04
    Let f(x)=x2+1x−1f(x) = \dfrac{x^{2} + 1}{x - 1}. Find the xx-coordinate of its local minimum.

    Curve sketchingChallenging

Start the set