Tests for determining when a function is increasing or decreasing

It is evident from Figure 8.7 that at a point where a function

$$y = f(x)$$

is increasing, the tangent in general makes an acute angle with the $x$-axis; hence

slope $= \tan\, \tau = \frac{dy}{dx} = f'(x) =$ a positive number.

Similarly, at a point where a function is decreasing, the tangent in general makes an obtuse angle with the $x$-axis; therefore^8.3

slope $= \tan\, \tau = \frac{dy}{dx} = f'(x) =$ a negative number.

In order, then, that the function shall change from an increasing to a decreasing function, or vice versa, it is a necessary and sufficient condition that the first derivative shall change sign. But this can only happen for a continuous derivative by passing through the value zero. Thus in Figure 8.7 as we pass along the curve the derivative (= slope) changes sign at the points where $x=1$ and $x = 2$. In general, then, we have at ``turning points,''

$$\frac{dy}{dx} = f'(x) = 0.$$

A value of $y = f(x)$ satisfying this condition is called a critical point of the function $f(x)$. The derivative is continuous in nearly all our important applications, but it is interesting to note the case when the derivative (= slope) changes sign by passing through^8.4 $\infty$. This would evidently happen at the points one a curve where the tangents (and curve) are perpendicular to the $x$-axis. At such exceptional critical points

$$\frac{dy}{dx} = f'(x) = \inf;$$

or, what amounts to the same thing,

$$\frac{1}{f'(x)} = 0.$$