Critical point
A value of x where f'(x) = 0 or f'(x) does not exist. Local maxima and minima can only happen at critical points.
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Analogy
The top of a hill and the bottom of a valley are both flat for a moment.
Example
f'(x) = 3x^2 - 3.Set to 0: x^2 = 1.Critical points at x = -1 and x = 1.f(-1) = 2, f(1) = -2.
In practice
Finding critical points is step one when searching for the highest or lowest value of a function.
Practice
Flashcards
- How do you find critical points?Solve f'(x) = 0 and find where f'(x) does not exist.
Interview questions
- 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
Is every critical point a maximum or minimum?
No. A critical point is only a candidate. For f(x) = x cubed, the derivative 3x squared is zero at x = 0, but the function keeps rising through that point. You need a further test, such as the sign of f' around it.
Why do maxima and minima occur at critical points?
At a smooth peak or valley the tangent line is flat, so the slope is zero. If the slope were positive or negative, you could move a little way and get a higher or lower value. Corners are the other possibility, where the derivative does not exist.