Understanding Function Behavior with First and Second Derivatives

1. What does say about f ? Increasing/decreasing test • If on an interval I, then f is increasing on I. • If on an interval I, then f is decreasing on I. • Proof. Use Lagrange’s mean value theorem.

2. Example • Ex. Find where the function is increasing and where it is decreasing. • Sol. Since when x¹0, and f is not differentiable at 0, we know f is increasing on (-1,0), (2,+1); decreasing on (0,2).

3. Example • Ex. Find the intervals on which is increasing or decreasing. • Sol. increasing on decreasing on • Ex. Prove that when • Sol. Let f(x)=sinx+tanx-2x, x2I =(0,p/2). Then f increasing on I, and f(x)>f(0)=0 on I.

4. Example • Ex. Show that is decreasing on (0,1). • Sol. • Ex. Prove that when x>0. • Sol.

5. The first derivative test • A critical number may not be a maximum/minimum point. • The first derivative test tells us whether a critical number is a maximum/minimum point or not: • If changes from positive to negative at c, then maximum • If changes from negative to positive at c, then minimum • If does not change sign at c, then no maximum/minimum • This explains why has no maximum/minimum at 0.

6. Example • Ex. Find all the local maximum and minimum values of the function • Sol. All critical numbers are: Using the first derivative test, we know: is local maximum point, are local minimum points

7. Convex and concave • Definition If the graph of f lies above all of its tangents on an interval I, then it is called convex (concave upward) on I; if the graph of f lies below all of its tangents on I, it is called concave (concave downward) on I. The property of convex and concave is called convexity (concavity). • Definition A point on the graph of f is called an inflectionpoint if f is continuous and changes its convexity. • The convexity of a function depends on second derivative.

8. Convexity test • If for all x in I, then f is convex on I. • If for all x in I, then f is concave on I. • Ex. Find the intervals on which is convex or concave and all inflection points. • Sol. By convexity test,f convex on and concave on and the inflection points are

9. The second derivative test • The second derivative can help determine whether a critical number is a local maximum or minimum point. • The second derivative test • If then f has a local minimum at c • If then f has a local maximum at c • Ex. Find the local maximum and minimum points of • Sol. Local maximum local minimum

10. Before sketching a graph • Using derivative to find the global and local maximum and minimum values, and locate critical numbers • Using derivative to find convexity and locate inflection points • Using derivative to find intervals on which the function is increasing or decreasing • Find domain, intercepts, symmetry, periodicity and asymptotes

11. Asymptotes • Horizontal asymptotes: if then y=L is a horizontal asymptote of the curve y=f(x) • Vertical asymptotes: if then x=a is a vertical asymptote of the curve y=f(x) • Slant asymptotes: if or then y=mx+b is a slant asymptote of the curve y=f(x)

12. Slant asymptote • Since to find slant asymptotes, we first investigate the limit if it exists, then and y=mx+b is a slant asymptote.

13. Homework 9 • Section 4.3: 14, 16, 17, 47, 49, 70, 74