- Topic
- Curve sketching
- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
Curve sketching practice question 26
A function is even, and its graph for is shown. Counting both sides of the -axis, how many local extrema (local maxima and local minima together) does have?
Even means : the left half is the mirror image of the drawn half in the -axis. Sketch it in.
Look closely at . The drawn half leaves climbing, so what does its mirror image do as it arrives at from the left?
A whole number.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Complete the graph by reflecting the drawn half in the y-axis, then follow the whole graph from left to right and mark every point where it switches between rising and falling.
Working
Read from the drawn half.
Reflecting in the y-axis keeps maxima as maxima and minima as minima.
The graph falls into (0, 1) from the left and climbs out of it to the right: a local minimum at a corner, where f′(0) does not exist.
Two local maxima (x = ±1) and three local minima (x = −3, 0, 3).
Answer
The answer is 5.
Check
f(0.1) and f(−0.1) are both about 1.28, above f(0) = 1, so the corner at 0 is a local minimum, alongside the turning points at x = ±1 and ±3.
Where this goes wrong
Missing the corner at x = 0. When the right half of an even function leaves the y-axis with a nonzero slope, the full graph has a V there, and the bottom of a V is a local minimum even though it has no horizontal tangent.
Leans on
Even and odd functions, from Functions, graphs and inverses
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