Hold on, if the first derivative is zero, the tangent to the function is a horizontal line (slope = zero) at that point or a horizontal plane, in the case of a two-parameter function. But consider the two dimensional case first. If the second derivative is zero, it means that the rate of change in the slope is zero. That occurs at an "inflection point" where the slope either stops increasing and starts decreasing, or vice versa. For example, the second derivative of the parabola f(x) = x^2 is just the number 2. It doesn't have any zeroes, so the slope continues to increase (become more positive as x increases from negative infinity to positive infinity. But the function f(x) = x^3 has a second derivative of 6x, and will be zero when x = 0. If you graph this and look at it you will see that as x increases from left to right, the slope is decreasing in the third quadrant until the point x=0, then increasing in the first quadrant. At the exact point where the curvature chances from "concave down" to "Concave up" there is an inflection point where the instantaneous rate of change of the slope is zero,

In the three dimensional case, the analog to an inflection point is indeed a "saddle point".

To test, evaluate the function for x plus or minus some small number, and the same for y. If the two values for x are less than the original value, the point is a maximum, ditto for y. If both x and y show a maximum, the point is the top of a "hill". If both a minimum, the bottom of a "hole." If one shows a maximum, the other a minimum, it is a saddle point.

I don't think your textbook actually says what you quoted. It probably says that a point where the first derivative will be a maximum if the second derivative is negative at that same point. Is that helpful?