Answer
The graph of the tangent function has a smallest slope but does not have a largest slope. The slope is also never negative.
Work Step by Step
The derivative of $\tan x$ is $\sec^2x$.
The graphs of $y=\tan x$ and $y=\sec^2x$ are enclosed below.
To find out all information of the maximum, minimum or the signs of the slopes to the tangent function, we can look at the behavior of the graph of its derivative, $y=\sec^2x$.
- To find out whether the graph of the tangent function has a smallest slope or not, we look for the minimum value in the graph of $y=\sec^2x$. The derivative graph does have its minimum value at $x=0$.
- To find out whether the graph of the tangent function has a largest slope or not, we look for the maximum value in the graph of $y=\sec^2x$. The derivative graph does not present any maximum values: it heads for infinity both $x$ gets smaller and larger.
- Is the slope of the tangent function ever negative? No. Because the graph of $y=\sec^2x$ lies completely on the positive side of the $y-$axis. The reason for this is simple: $\sec^2x\ge0$ for all possible $x$.
