Question

How do you use the first and second derivatives to sketch `-2(x-2)(x+3)(x+4)`?

Answer 1

Use the first derivative to find both the slopes on given intervals of the function, and critical points of the function. Use the second derivative to determine where the function is concave up or concave down.

Explanation:

In order to plot this function, we will use the following bits of information:

(1) The zeroes of the function: At these points, f(x) =0. Thanks to the equation being factored, this is easily determined. `f(x) = 0` at `x=2, x=-3, x=-4`

(2) Whether the function is increasing or decreasing at its zeroes. To find this, we will calculate the derivative of the function, and then plug in each zero. If the derivative is positive, the function is increasing at that zero; if negative, it is decreasing; and if the derivative is 0, the function is neither increasing or decreasing.

(3) The critical points of the function. This will be determined by calculating the derivative of the function, and finding the zeroes of the derivative. These zeroes are critical points for the function, i.e. points where the function is neither increasing nor decreasing. These may take the form of plateaus, local minima, or local maxima.

(4) The type of critical point. Look at a small interval around a given critical point `c`, say `[c-epsilon, c+epsilon]` where epsilon is arbitrarily small. If the derivative goes from negative to positive along the interval, the point is a local minimum, meaning that `f(c) < f(c+-epsilon)`. If the derivative goes from positive to negative, the point is a local maximum, meaning `f(c) > f(c+-epsilon)`. If the derivative retains the same sign, it is a plateau; the function does not switch direction.

(5) Concavity. To find the concavity, we need the second derivative. Where the second derivative is positive, the function is convex, aka concave up, meaning that for any two points on the curve (i.e. `(a, y_1) (b, y_2)`), the line connecting these two points is above the graph of the function in that area. When the second derivative is negative, the function is concave downward, and the line connecting the points is below the graph.