1. Math 1304 Calculus I
4.03 – Curve Shape

2. Increasing/Decreasing
Theorem:
(a) If f’(x) > 0 on an interval, then f is increasing on that interval.
(b) If f’(x) < 0 on an interval, then f is decreasing on that interval.
Proof: Use MVT.
Let x1 < x2 be different numbers in the interval.
Then there is a c in (x1, x2) such that f’(c) = (f(x2)-f(x1))/(x2-x1).
But f’(c)>0.
So (f(x2)-f(x1))/(x2-x1) > 0.
So f(x2)-f(x1) > 0. Thus f(x2) > f(x1).
(b) Left to students.

3. First Derivative Test
First Derivative Test: Suppose that c is a critical number of a continuous function f.
If f’ changes from positive to negative at c, then f has a local maximum at c.
If f’ changes from negative to positive at c, then f has a local minimum at c.
If f’ does not change sign at c, then f has no local maximum or minimum at c.

4. Concavity
Definition: If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on an interval I, then it is called concave downward on I.
Concavity Test:
If f’’(x) >0 for all x in I, then the graph of f is concave upward on I.
If f’’(x) < 0 for all x in I, then the graph of f is concave downward on I.

5. Inflection Point
Definition: A point P on a curve y = f(x) is called an inflection point if f is continous there and the curve changes concavity there.

6. Second Derivative Test
Second derivative test: Suppose that f’’ is continuous near c.
If f’(c) = 0 and f’’(x) > 0, then f has a local minimum at c.
If f’(c) = 0 and f’’(x) < 0, then f has a local maximum at c.