- Point moving on parabola from problem set
- Section 4.1
- Maximum and minimum points and how to find them
- Definitions
- Absolute (related to domain): maximum (or minimum) value function has over whole domain
- Local
- Book (maximum): f(x) ≤ f(c) for all x in some open interval around c
- Colloquial: (x,y) point where y is bigger (smaller) than neighboring y values
- Find maxima/minima by looking for places where derivative = 0
- Functions can have multiple local maxima and minima; some even
have multiple occurrences of absolute maxima or minima
- Examples
- Where is the maximum value of y = 4 - x2?
- Expect maximum at (0,4)
- dy/dx = -2x
- 0 = -2x implies x = 0
- Plug x=0 in to 4 - x2 = 4
- Where is/are maxima/minima of y = x3 - 3x for -5 ≤ x ≤ 5?
- At x = -5 (minimum) and x = 5 (maximum)
- Not where derivative = 0: those points (x = -1 and x = 1) are local maximum and minimum
- Moral: maxima/minima can appear at endpoints of closed
interval regardless of derivative there
- Where is the maximum value of y = 4 - x2?
- More extreme value examples
- Mean Value Theorem
- Read section 4.2