Second derivative test
At a critical point, if f'' is positive the curve bends upward (a minimum). If f'' is negative it bends downward (a maximum).
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Analogy
A bowl holds water (minimum), an upside-down bowl sheds it (maximum).
Example
f''(-1) = -6, negative: maximum.f''(1) = 6, positive: minimum.f''(0) = 0 and the bend flips.So x = 0 is an inflection point.
In practice
Often quicker than checking signs on both sides, and f'' also tells you where the curve changes bend.
Practice
Flashcards
- If f'(c) = 0 and f''(c) < 0, what is c?A local maximum.
Interview questions
- A rectangle has perimeter 100 m. Find its dimensions for maximum area and justify that it is a maximum.
- Let sides be x and 50 - x
- A = 50x - x^2
- A' = 50 - 2x = 0 gives x = 25
- A'' = -2 < 0 so a maximum
- Area is 625 m^2, a square
- Find and classify the critical points of f(x) = x^3 - 3x.
- f'(x) = 3x^2 - 3 = 0 gives x = -1 and 1
- f''(x) = 6x
- f''(-1) < 0 so local maximum, value 2
- f''(1) > 0 so local minimum, value -2
Questions people ask
What does the second derivative tell me?
It tells you how the slope is changing. A positive f'' means the slope is increasing, so the graph is concave up like a cup. A negative f'' means the slope is falling and the graph is concave down like a cap.
What if f'' equals zero at a critical point?
The test gives no answer. The point could be a maximum, minimum or neither. Go back to the first derivative test. For x to the fourth, x cubed and negative x to the fourth, f'' is zero at 0, yet the behaviour differs in each case.