# 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