2 Section OutlineThe Second Derivative TestGraphs With Asymptotes
3 Second Derivative Test EXAMPLESketch the graph ofSOLUTIONWe haveWe set and solve for x.(critical value)
4 Second Derivative Test CONTINUEDSince , we know nothing about the graph at x = 2. However, the test for inflection points suggests that we have an inflection point at x = 2. First, let’s verify that we indeed have an inflection point at x = 2. If this proves to be not the case, we would use a similar method (using the first derivative) to see if we have a relative extremum at x = 2.We will look at the left- and right-hand limits of and see if the values of the second derivative are oppositely signed as x approaches 2.x x - 12x x - 12
5 Second Derivative Test CONTINUEDFrom the values of in the table, we find that, as x approaches 2, the second derivative not only has a limit of 0, but the values of the function to the left of x = 2, remain negative. Also, the values of the function to the right of x = 2 remain positive. Therefore, by definition, f (x) has an inflection point at x = 2 (since the graph of f (x) is concave down to the left of x = 2 and concave up to the right of x = 2).Notice, x = 2 was the only candidate for generating a relative extremum. Therefore, there are no relative extrema. We will now find the y-coordinate for the inflection point.So, the only inflection point is at (2, 3).
6 Second Derivative Test CONTINUEDNow we will look for intercepts. Let’s first look for a y-intercept by evaluating f (0).So, we have a y-intercept at (0, -5). To find any x-intercepts, we replace f (x) with 0.Since this equation does not factor, and the quadratic formula cannot help us either, we attempt to use the Rational Roots Theorem from algebra. In doing so we find that there are no rational roots (x-intercepts). So, if there is an x-intercept, it will be an irrational number. Below, we show some of the work employed in estimating the x-intercept.
7 Second Derivative Test CONTINUEDx f (x)Notice that the y-values corresponding to x = 0.54 and x = 0.55 are below the x-axis and the y-values corresponding to x = 0.56 and x = 0.57 are above the x-axis. Therefore, in between x = 0.55 and x = 0.56, there must be an x-intercept. For the sake of brevity, we’ll just take x = 0.56 for our x-intercept since, out of the four x-values above, it has the y-value closest to zero. Therefore, the point of our x-intercept is (0.56, 0).Now we will sketch a graph of the function.
8 Second Derivative Test CONTINUED(0.56, 0)(2, 3)(0, 5)
9 f (x) yields information about where things are on a graph. yields information about slope on a graph.yields information about concavity on a graph.
10 Graphs With Asymptotes EXAMPLESketch the graph ofSOLUTIONWe haveWe set and solve for x.(critical values)
11 Graphs With Asymptotes CONTINUED(We exclude the case x = -2 since this is outside our specified domain.) The graph has a horizontal tangent at (2, f (2)) = (2, 13). Since , the graph is concave up at x = 2 and, by the second derivative test, (2, 13) is a relative minimum. In fact, for all positive x, and therefore the graph is concave up at all points.Before sketching the graph, notice that as x approaches zero the term 12/x in the formula for f (x) is dominant. That is, this term becomes arbitrarily large, whereas the terms 3x + 1 contribute a diminishing proportion to the function value as x approaches 0. Thus f (x) has the y-axis as an asymptote. For large values of x, the term 3x is dominant. The value of f (x) is only slightly larger than 3x since the terms 12/x + 1 has decreasing significance as x becomes arbitrarily large; that is, the graph of f (x) is slightly above the graph of y = 3x As x increases, the graph of f (x) has the line y = 3x + 1 as an asymptote.
12 Graphs With Asymptotes CONTINUEDy = 3x + 1(2, 13)