MATH 1500: Maxima and Minima

What is Fermat’s theorem?

If x is a local extreme point, then either

• f is differentiable at x and f’(x) = 0 (i.e. x is a critical point); or

• f is not differentiable at x (i.e. x is a singular point)

What is the extreme-value theorem?

If f(x) = continuous on a closed interval [a,b], then f(x) has both a minimum and maximum on the interval.

How do you find the local and absolute extrema of a function?

1. Find f’(x)

2. Find critical numbers (“roots”) and check if in domain. As well as checking if f(x) exists while f’(x) DNE (i.e. check for differentiability) using left-hand and right-hand when needed.

3. Make number line and make sign chart with intervals, factors, sign of f’(x), and increasing-/decreasing-ness of f(x) to predict local extrema when sign changes (i.e. FDT)

4. Make chart for x values (endpoints/singular points, roots/critical numbers) and y values (output of x).

5. Determine if abs/local min/max points and values based on highness and lowness.

Why is a f(x) not continuous at a point if the limit doesn’t exist?

> Continuity = no gaps at that point

> If limit DNE, the left and right values don’t meet → there’s a gap → not continuous.

Why does lim f(x) ≠ f(a) as x→a mean not continuous?

> Continuity means the graph reaches the same point it approaches.

> If the approached value and actual value differ, there’s a break/gap.

How to check for continuity at some point a?

1. Check if f(a) exists. If undefined = not continuous.

2. Check if lim f(x) exists as x→a (for suspicious cases: check left = right). If lim DNE = not continuous.

3. Check if lim f(x) = f(a) as x→a. If not equal = then not continuous.

Why does not continuous → not differentiable?

> Differentiability = smooth slope

> If graph has break/jump/gap, there’s no smooth tangent line to take a derivative from.

Why compare left and right derivatives for suspicious functions?

> b/c the slope might change suddenly.

> Differentiability needs ONE smooth matching slope from both sides.

Why can smooth functions just use f′(a)?

> b/c smooth f(x) have matching slopes naturally

> If f’(x) exists, then the graph is smooth there.

How do you check for differentiability?

1. Identify point.

2. Check continuity. If not continuous, not differentiable.

3. For suspicious functions, compare left and right derivatives. If f’_-(a) = f’₊(a) = differentiable. (for normal, smooth functions, just find f’(a). If it exists = differentiable.