In today's video, we formally discuss concavity, which is a property of curves with the slopes of tangent lines are either increasing or decreasing, leading to notions of concave up and concave down, and points of inflection where concavity changes either from concave up to down or from concave down to up, which will also point to the property that the tangent lines pass through the curve. We begin by discussing concavity. Think of the property concave up of a curve as corresponding to the curve being in a bowl-shaped up configuration. If you look at miniature tangent lines to the curve as you pass from left to right, you'll notice that the slopes change from being very steep and negative, to shallow and negative, to being zero as you move through the turning point, to being shallow and positive, to being steep and positive. The slopes of the tangent lines are increasing. This corresponds to the derivative of the rule of the original function being itself an increasing function. So, its derivative, the second derivative of the original function denoted by y double dashed, should be positive. Now, consider the opposite property concave down, which corresponds to the curve being in a bowl-shaped down configuration. Now, if you look at miniature tangent lines to the curve as you pass from left to right, you'll notice that the slopes change from being very steep and positive, to shallow and positive, to being zero as you move through the turning points, to being shallow and negative, to being steep and negative. The slopes of the tangent lines are decreasing. This corresponds to the derivative of the rule of the original function being itself a decreasing function. So, the second derivative of the original function, y double dash, should be negative. A point of inflection or we simply say an inflection, is a point where the concavity changes either from being concave up to being concave down, or from concave down to concave up. We can picture the first case in a couple of ways transitioning from bowl-shaped up to bowl-shaped down, and the moments of transition are called inflections. You can watch the way miniature tangent lines to the curve behaves as you pass from left to right for the first curve displayed here, and for the second curve. You might notice that the tangent line crosses the curve at the point of inflection. The slope of the tangent line which is the derivative denoted by y dash, reaches a maximum at that moment of transition at the inflection point. So, its derivative denoted by y double dash must become zero. In summary, for the case where concavity changes from concave up to down, the derivative reaches a maximum and the second derivative is zero. Consider the second type of transition from concave down to concave up. We can picture that in a couple of ways transitioning from bowl-shaped down to bowl-shape up. Note the moments of transition, the inflections. Now watch the way the miniature tangent lines behave as you pass from left to right. For the first curve and for the second curve, and again, the tangent line crosses the curve at the point of inflection. This time the derivative y dashed reaches a minimum at that moment of transition at the inflection, and again, the second derivative y double dashed must become zero. In summary, for the case where concavity changes from concave down to up, the derivative reaches a minimum and the second derivative is zero. So, we've considered variations in the way concavity can change and the effects they have on the derivative and second derivative. The upshot of this is that changes in concavity may be detected by looking at the changes in the sign of the second derivative, and all of this information will be captured in its sign diagram. To see how this works, let's sketch the curve y equals 9x minus x cubed. We begin by noticing that the rule factorizes as x times three plus six times three minus x, which will be useful in a moment for detecting x intercepts. The derivative y dashed equals nine minus 3x squared which factorizes three times root three plus x times root three minus x. The second derivative y double dash is simply minus 6x. Putting these together, we can start to build sign diagrams and we'll do this for the first and second derivatives. In the case of y dash, we say that the important inputs for x are plus minus root three where the derivative is zero, and we get a pattern of signs, minus plus minus as we move from left to right. So, the curve is decreasing then increasing then decreasing again. In the case of y double dashed, the important input for x is zero where the second derivative is also zero, and the pattern of signs is plus minus as we move from left to right. So, the curve must be concave up to the left of zero indicated by the smiley face symbol, and concave down to the right of zero indicated by the sad face symbol. All of this information is noted here to the side and we can list the main features of the curve. The y-intercept is just zero, the x-intercepts will be 0, 3 and minus three. From the sign diagram for the derivative, there is a local minimum occurring for x equal to minus root three and a local maximum when x equals plus root three. From the sign diagram for the second derivative, there's an inflection at the origin. Now we can sketch the curve with confidence. Here are the important points on the x-axis and we're fitting a curve which is concave up to the left of zero, concave down to the right, with a local minimum at x equal to negative root three and a local maximum at x equal to plus root three. The local maximum and minimum values, you can quickly work out from the rule for the function. And they become plus or minus six times the square root of three, and the origin is the points of inflection where the concavity changes. This completes the sketch of the curve y equals 9x minus x cubed. We'll finish with an example of a cusp and sketch the curve y equals x to the power of two thirds. We begin by differentiating y, which becomes two thirds of x to the minus a third, remembering the rule that the exponent comes down to the front and you subtract one to get the new exponent, which we can rewrite as two over three times x to the one third. The sign diagram for the derivative involves one important point for x, namely x equal to zero, where the derivative is not defined because otherwise we'd be trying to divide by zero in the rule for y dashed. The pattern of signs is simply negative, positive as the sign of y dashed agrees with sign of x. So, the curve is decreasing to the left of zero and increasing to the right. This means that there is a global minimum at x equals zero, noting that the curve is continuous even though the derivative is undefined at x equals zero. We can say more about the behavior of the derivative near the origin using limit notation. The limit of y dashed as x approaches zero from below is a limit of two divided by three times the cube root of x, which becomes arbitrarily large and negative, and we use the minus infinity notation to capture this. On the other hand, the limit of y dashed as x approaches zero from above becomes plus infinity capturing the idea that the derivative gets arbitrarily large and positive. These two limits summarize the behavior of the slopes and imply that the tangent lines themselves become more and more vertical as x approaches zero from one side or the other. Let's continue differentiating. The second derivative becomes two-thirds of minus a third times x to the minus four-thirds, just using the rule for differentiating a power of x again, which may be rewritten as negative two divided by nine times the square of x to the two thirds. We're emphasizing the square in the denominator as this tells us that it's always positive. So, the entire fraction is always negative provided x is not equal to zero. We can now build the sign diagram for y double dash, which again has zero for an important point for x and y double dash is undefined. Now, the pattern of sign is just negative, negative. So, we have the concave down sad face symbol in both cases. In summary, we have the derivative and second derivative and their respective sign diagrams, and the contrasting unbounded behaviors of the derivative as x approaches zero from each side. We can now draw the graph that matches all of this information producing a continuous curve with a sharp point, a cusp at the origin, which also coincides with the global minimum value of y equals zero. This completes the sketch of the curve. In today's video, we discussed notions of concavity of a curve, including the properties of being concave up and concave down, and points of inflection where concavity changes either from concave up to down or from concave down to up. We illustrated the ideas by working through a couple of contrasting examples, including a cusp and exploited information provided by the sign diagrams to the first and second derivatives. Please read the notes and when you're ready, please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
