Before Quiz 2

Ignoring the points where f′ does not exist

The habit: Look for where f′ is undefined as well as where it is zero, since a cusp or a corner can be a local extremum.

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

Practise against it: 3 questions

What it looks like

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

Standard3 marks · 5 min

Let $f(x) = x^{2/3}(x - 5)$ , defined for every real $x$ . Which statement about its critical points, singular points and local extrema is correct?

With the slip: Its only critical point is x = 2, a local minimum, and f has no other local extremum.
Worked through: x = 0 is a singular point, where f has a local maximum (a cusp), and x = 2 is a critical point, where f has a local minimum.

f′ = 0 finds x = 2 only. But f′(x) = 5(x − 2)/(3∛x) is undefined at x = 0, where f is continuous: check the sign of f′ on each side of 0.

See the worked solution

Where it comes up

Differentiation rules1 question
Curve sketching2 questions
Optimization in two variables and Lagrange multipliers1 question

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
The graph of $f(x, y) = \sqrt{(x - 1)^{2} + (y + 2)^{2}}$ is a cone with its tip on the $xy$ -plane. Which statement about the critical points of $f$ is correct?
Optimization in two variables and Lagrange multipliersIntroductory
02
Let $f(x) = \dfrac{\sqrt{x}}{x^{2} - 4}$ . For which $x$ does $f'(x)$ exist?
Differentiation rulesStandard
03
A function $f$ is even, and its graph for $x \geq 0$ is shown. Counting both sides of the $y$ -axis, how many local extrema (local maxima and local minima together) does $f$ have?
Curve sketchingChallenging

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