Derivatives and the shape of the graph

The first derivative test.

  • 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

  • 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

  • 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