Before the final

Answering where instead of what

The habit: Read whether the question wants the x at which the extreme occurs or the value there, and give that one.

15 practice questions are built to catch it, in 8 topics.

See it fail in a figure: Where, and what, in the optimization in two variables and Lagrange multipliers lesson

Practise against it: 6 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 global minimum value of $f(x) = x^{3} - 3x + 1$ on the closed interval $[0, 2]$ .

With the slip: $\displaystyle 1$
Worked through: $\displaystyle -1$

x = 1 is where the minimum occurs. The question wants the minimum value: put x = 1 into f, and compare with f(0) = 1 and f(2) = 3.

See the worked solution

Where it comes up

Differentiation rules1 question
The chain rule and logarithms1 question
Implicit differentiation and inverse trigonometric functions2 questions
Curve sketching1 question
Optimization5 questions
Slope fields and Euler’s method1 question
Logistic growth and phase lines2 questions
Optimization in two variables and Lagrange multipliers2 questions

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.

01
Find the coordinates of the local minimum of $h(x) = x^{3} - 27x + 5$ .
OptimizationIntroductory
02
The curve $y = x^{3} - 3x^{2} + 5$ has a tangent line parallel to $y = 9x + 1$ at exactly one point with $x > 0$ . Find the $y$ -intercept of that tangent line.
Differentiation rulesStandard
03
The graph of $y = \arctan x$ has exactly two tangent lines parallel to the line $y = \tfrac{1}{2}x$ . Find the $y$ -intercept of the one that touches the graph at a point with $x < 0$ .
Implicit differentiation and inverse trigonometric functionsStandard
04
Find the local maximum value of $f(x) = x^{3} - 3x^{2} - 9x + 5$ .
Curve sketchingStandard
05
The curve $y = x\,e^{-x^{2}/8}$ has exactly two points where the tangent line is horizontal. Find the $y$ -coordinate of the higher of the two.
The chain rule and logarithmsChallenging
06
The curve $x^{2} + xy + y^{2} = 3$ has exactly two points at which its tangent line is horizontal. Find the $y$ -coordinate of the higher of those two points.
Implicit differentiation and inverse trigonometric functionsChallenging