(a) the intervals on which f increases and the intervals on...

$(a)$ the intervals on which $f$ increases and the intervals on which $f$ decreases; (b) the local maxima and the local minima; (c) the intervals on which the graph is concave up and the intervals on which the graph is concave down; (d) the points of inflection. Use this information to sketch the graph of $f.$ $$f(x)=x^{1 / 3}(x-6)^{2 / 3}$$

Key Concepts

Increasing and Decreasing Functions

This concept involves determining the intervals on the domain of a function where it is rising (increasing) and where it is falling (decreasing). The analysis is typically conducted using the first derivative of the function; by finding where the derivative is positive or negative, one can detect the behavior of the function. Understanding this concept is crucial for identifying local peaks and valleys as well as for sketching an accurate graph of the function.

Local Extrema

Local extrema refer to the local maximum and minimum values of a function—points on the graph where the function attains a relatively highest or lowest value, respectively. Utilizing the first derivative test, one identifies potential candidates for these extrema by analyzing where the derivative is zero or undefined, and by checking the sign change around these points. These are essential markers when sketching the graph since they indicate the turning points of the function.

Concavity

Concavity describes the curvature of the function's graph and can be determined by the sign of the second derivative. When the second derivative is positive over an interval, the graph is concave up (curving upwards), and when it is negative, the graph is concave down (curving downwards). Recognizing concavity helps in comprehending the overall shape of the graph and is instrumental in predicting the behavior of the function between critical points.

Inflection Points

Inflection points are specific points on the graph where the concavity changes from upward to downward or vice versa. These points occur where the second derivative is zero or undefined, provided there is an actual change in concavity. Detecting inflection points is important as they indicate a change in the trend of the graph, and knowing their location aids in understanding the complete behavior of the function.

Graph Sketching Using Derivatives

Combining the concepts of increasing/decreasing intervals, local extrema, concavity, and inflection points allows one to sketch the graph of a function accurately. By first finding the critical points via the first derivative and determining the curvature through the second derivative, one can piece together the function’s behavior across its domain. This methodical approach is key for visualizing how the function behaves, even in cases where the explicit equation may initially seem complex.

Transcript

00:02 And this problem, we have a bit of an interesting function.

00:07 I'm not sure how you really describe it very well, but it is the cube root of x times the cube root of x minus six squared.

00:20 So that's clearly zero when x is zero or six.

00:23 So we have these two points here.

00:28 Now, taking a derivative, again using the product rule here, and simplifying, we get x minus 2 divided by x minus 6 to the 1 third times x to the 2 thirds.

00:44 And that's 0 when x equals 2.

00:47 So we have a horizontal tangent there.

00:53 Now we also have a, you can also see that here it goes to infinity when x goes to minus 6 and also when x goes to 0.

01:05 So we need to kind of be careful of those points 6 and 0 also.

01:11 Taking a second derivative, we get 8 all of over x minus 6 to the 4 thirds times x to the 5 thirds.

01:19 Now that's never equal to 0, but it is equal to infinity at minus 6.

01:25 So we need to be careful about that point also.

01:28 So we take a look at this, and we see here, this changes sign between it's an increasing function over here.

01:38 It's decreasing, oh sorry, it's decreasing here, and it's increasing again here.

01:43 And this is curvature up, and this is down.

01:48 So what we have then is if we look at this, when x is less than 2, it's positive.

01:58 Because this is negative and this is negative.

02:02 Oh, it's certain less than 0, and also when x is less than 0, it's negative.

02:08 It's right, it's positive...

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