SectionΒ 3.1 and SectionΒ 3.2 introduce connections between derivatives and their functions including possible locations of relative extrema. In this section examples of using these traits to find and indentify the types.
The derivative will be zero only where the numerator is zero which occurs at \(x=1,2,3\text{.}\) The derivative will be undefined where the denominator is zero which occur at \(x=1,3\text{.}\) Thus possible locations for relative extrema are \(x=1,2,3\text{.}\)
After using the derivative to determine where relative extrema might occur we can use values of the function. For this test we evaluate the function at the critical values and left and right of the critical values.
After using the derivative to determine where relative extrema might occur we can use values of the derivative to determine where the function is increasing and decreasing. For this test we evaluate the derivative left and right of each critical value.
After using the derivative to determine where relative extrema might occur we can use values of the second derivative to determine which are maxima and which are minima. For this test we evaluate the second derivative at the critical values found.
The second derivative only helps us at \(t=2\text{.}\) Because the curve is concave down, this is a relative maximum. The other values are undefined (cusps), so this test is not helpful.