- Increasing/decreasing test.
- Consider the graph of a differentiable function \(C:y=f(x)\) on an interval \(I\).
- Recall that the derivative \(f'(x=c)\) is the slope of the tangent line to \(C\) at \(x=c\).
- Therefore, the derivative can tell us if the function is increasing or decreasing.
*Increasing/decreasing test.*- If \(f'(x)>0\) on \(I\), then \(f\) is strictly monotonously increasing on \(I\).
- If \(f'(x)<0\) on \(I\), then \(f\) is strictly monotonously decreasing on \(I\).
- Note that it is when \(f\) changes its monotonicity (goes from increasing to decreasing or vice versa) that it has a local extreme value.
- Therefore, the first derivative \(f'\) can help us decide if \(f\) has a local extreme value at a given critical number, and if so, a maximum or a minimum.

- The first derivative test.
- Let \(f\) be a function, and let \(c\) be a critical number of \(f\).
- If \(f'\) changes from positive to negative at \(x=c\), then \(f(x=c)\) is a local maximum.
- If \(f'\) changes from negative to positive at \(x=c\), then \(f(x=c)\) is a local minimum.
- If \(f'\) has the same nonzero sign to the left and right of \(x=c\), then \(f(x=c)\) is not a local extreme value.
- Exercises. 4.3: 10ab, 12ab, 14ab

- Concavity test.
- So far, we have only talked about how to decide if a function is increasing or decreasing.
- But there's many ways a functions can be increasing. For example, both \(e^x\) and \(\ln x\) are strictly monotonously increasing functions, although \(e^x\) is increasing really fast, and \(\ln x\) is increasing really slow.
- One thing we can check is the rate of change of the derivative. In other words, the acceleration.
- Note that this notion is independent of the monotonicity of \(f\).
*Definition*. If the graph of \(f\) lies above all its tangents on an interval \(I\), then it is called*concave upward*on \(I\). If the graph of \(f\) lies below all its tangents on \(I\), 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 \(I\).
*Definition*. A point \(P\) on a curve \(y=f(x)\) is called an*inflection point*, if \(f\) is continuous there, and the graph changes from concave upward to concave downward, or vice versa.

- The second derivative test.
- Contemplate the graph of \(f\) around a local minimum. For example, think \(y=x^2\). You can see that it needs to be concave upward.
- Similarly, around a local maximum, the graph of \(f\) needs to be concave downward.
*The second derivative test.*Suppose \(f\) is continuous near \(x=c\).- If \(f'(x=c)=0\) and \(f''(x=c)>0\), then \(f(x=c)\) is a local minimum.
- If \(f'(x=c)=0\) and \(f''(x=c)<0\), then \(f(x=c)\) is a local maximum.
- Exercises. 4.3: 10c, 12c, 14c, 22, 46, 50, 56, 68, 70, 72, 74