Critical Points — Where the Slope Is Zero
“At peaks and valleys, the slope is zero.”
The formula
f'(x) = 0How to read it: the x where the derivative (slope) is 0 — a candidate for a local max or min
- f'(x)
- — the derivative = the slope at that point
- = 0
- — slope is 0, the tangent is horizontal
- x
- — a candidate spot for a max or min
The hook
Finding a function's peaks and valleys is startlingly simple — 'look for where the slope is 0'.
In plain words
Set the derivative equal to 0 and solve; the x's you get are where the tangent goes flat — candidates for a max or min.
The intuition
Climb a hill and at the summit you're momentarily level. Same at the bottom of a valley. At the turning point between going up (slope +) and coming down (slope −), the slope passes exactly through 0.
How it's built
Solve f'(x)=0 to collect candidate x's. Then decide whether each is a peak or a valley using the sign of the slope on either side, or the second derivative.
Example
For f(x)=x²−4x+3, f'(x)=2x−4. Set it to 0 → x=2. There f(2)=4−8+3=−1 — the lowest point of this parabola.
Common mistake
Slope 0 doesn't always mean an extremum. f(x)=x³ has f'=0 at x=0, but it's neither a peak nor a valley — it just glides through (an inflection).
Where it's used
Max profit, min cost, shortest path — every optimization problem that hunts for the 'best value' starts by solving slope = 0.
Where it came from
Before differentiation was even formalized, Fermat planted the seed with the idea that 'change stalls near a maximum'.
Prerequisites
Quick check
At which x does f(x)=x²−6x have its extremum?
- 0
- 2
- 3✓
- 6
Practice
At which x does f(x)=x²−2x have its extremum?
Answer: 1
- f'(x)=2x−2
- Set to 0: 2x−2=0 → x=1
Takeaway: Differentiate, set to 0, solve for x.
At which x is f(x)=x²+6x+5 smallest?
Answer: -3
- f'(x)=2x+6
- Set to 0: 2x+6=0 → x=−3
Takeaway: For an upward parabola, slope 0 is the lowest point.
Find all critical points (where the slope is 0) of f(x)=x³−3x.
Answer: undefined
- f'(x)=3x²−3
- Set to 0: 3x²−3=0 → x²=1 → x=1, x=−1
Takeaway: There can be more than one critical point — collect every solution.