Use the guidelines of this section to make a complete graph of...

Question

Use the guidelines of this section to make a complete graph of f.

f(x) = 2 - 2x^2/3 + x^4/3

Accepted Answer

Hello. In this video, we are going to be graphing the function given in the problem. The function given to us is F of X equal to 6 minus 6 X 2/3, plus X raises the power of 8/3. Now, in order to graph this problem, there's a few pieces of information that we need to solve for. First, let's go ahead and discuss the domain of the function. Now, the function is in the form of a polynomial, and the convenient thing about this being a polynomial, is that we can conclude that the domain will be all real numbers, since any value of X will always work with polynomial. Next, let's go ahead and determine the Y intercept. In order to find the Y intercept, we are going to plug in 0 into our function. Plugging in 0 will cause any value of X to become zero, and that is going to leave us with 6. So what this means is that since the value of F of 0 is equal to 6, we have a Y intercept at 0.6. We will go ahead and label that on our graph. Next, let's go ahead and determine the symmetry of the function. In order to determine the symmetry, we will plug in the value negative X into our function. That is going to leave us with 6 minus 6 multiplied by negative X to the 2/3 power, plus X X raises the power of 8/3. Now, 2/3 and 8/3 are even powered functions or exponents. So what this means is that negative X is going to simplify to be positive X under an even exponent, and that is going to simplify F of negative X to be 6 minus 6 X rated to the 2/3, plus X raises to 8/3. This is equivalent to the original function, and since F of negative X is equal to F of X, that tells us that the function is going to be even. Since the function is even, that means that we have symmetry about the y axis. So, now that we know that we have symmetry about the y axis, we have a Y intercept in the domain, let's go ahead and determine the behaviors of the graph. Now, the convenient thing here is that since we know that this graph is equivalent or symmetric about the y axis, all we have to do is determine the behaviors of the graph from 0 to infinity. Then we can just mirror those behaviors about the y axis. So, let's go ahead and determine the potential maximum or minimum value on the right hand side of the function. So, in order to determine these values, we will go ahead and take the derivative of the function. We will use the power rule to take the derivative. The power rule will give us 0 minus 6 multiplied by 2/3, multiplied by X raise the power of negative 1/3. Plus 8/3 multiplied by X raises the power of 5/3. Simplifying the derivative will leave us with -4 divided by X 1/3 power, plus 8/3 multiplied by X rates to the 5/3 power. Now what we want to do is we want to go ahead and simplify the function. Or further yet, in order to identify the maximum or minimum values, we are going to find the values of X that cause the derivative to be zero. So, in order to do this, we will set our derivative equal to 0 and solve for X. We will start by adding 4 -4 or 4 divided by X3 power to both sides of the equation. That will leave us with 8/3 X rates to the power of 5/3 equal to 4 divided by X to the 1/3 power. We need to remove this X from the 1/3 power from the denominator. So we will go ahead and multiply both sides of the equation by X to the 1/3. Next, what that will leave us with is 8/3 multiplied by X rates to the 6/3 power equal to 4. The exponent of 6/3 was simplified to be 2, so we will have an X2 term on the left hand side of the equation. Finally, if we multiply both sides of the equation by the reciprocal of 8/3, we will be left with X2 is equal to 12 divided by 8. This will further simplify to be three halves. Finally, by taking the square root of both sides of this equation, we will have X's equal to positive and negative square root of three halves. So this is going to be the potential maximum or minimum values for the given graph, but what we will go ahead and do is use the first derivative test. We will go ahead and plot both of these values on a number line. We will have negative square root 3 halves and positive square root 3 halves. But there is one more value that we need to plot on this number line. Notice that if we take a look at the original derivative, we have a value of X that is in the denominator of a fraction. What this means is that this is a restriction of the of the derivative and therefore we have to plot this restriction on the number line to determine the behaviors of the graph. Now, what we will go ahead and do is we will go ahead and determine the behaviors between 0 and square root 3 divided by 2, and from square root 3 divided by 2 to positive infinity. The reason why we only determine these two behaviors is because we know the site, the graph is symmetric about the Y axis, therefore, the opposite end will just be mirrored. So what is a testing point that we can take between 0 and square root 3/2, that we can plug into the derivative? Well, that testing point is going to be positive one. When we plug in 1 into the derivative, F1 will give us the value of -4 divided by 3. What this tells us is that the graph is going to be decreasing in on the region from 0 to square root 3 divided by 2. And a testing point that we can take to the right of square root 3 divided by 2 is 2. When we plug in 2, F of 2 will have an approximate value of 5.29. But since this value is positive, the graph will be increasing in this region. So, what this also tells us is that because the graph is switching from decreasing to increasing, we have a local minimum. At the value of square root 3 divided by 2. Now what we want to go ahead and do is we want to determine the Y value for square root 3 divided by 2. If we plug in this value into the original function, we get F of square root 3 divided by 2, is approximately equal to 0.85, so just above the X axis. Let's go ahead and plot this point on our on our graph. So we have square root 3 divided by 2, which is roughly between 1 and 2. So we can estimate this here to be square root 3 divided by 2. 0.85. And again, since we have symmetry about the y axis, we will go ahead and mirror this point on negative square root 3 divided by 2. We will have the square root 3 divided by 2.0.85. No, We know that the graph is going to be decreasing between 0 and square root 3 divided by 2, and increasing from square root 3 divided by 2 to to infinity, but we need to check the concavity of the graph within these regions. So what we will go ahead and do is we will go ahead and calculate the second derivative of the function. The second derivative by the power rule will be F of X equal to 4 divided by 3 multiplied by X to the power of 4/3. Plus 40 divided by 9, multiplied by X raises the power of 2/3. We will go ahead and find the inflection points of the 2nd derivative, and we will find these points by setting the second derivative equal to 0. So we will go ahead and start by subtracting 40 divided by 9, multiplied by X to 2/3 to both sides of this equation. That will leave us with 4/3 X to the power of 4/3 equal to -40 divided by 9. Multiplied by X of the power of 2/3. Next, what we will go ahead and do is we will remove X to the power of 4/3 from the denominator by multiplying this entire equation by X 4/3. That will leave us with 4/3 equal to -40 divided by 9, multiplied by X rates the power of 6/3. And again, this is going to simplify to be 2, so we have an X2 term. Next, we will go ahead and multiply both sides of this equation by The reciprocal of 40 divided by 9, we will multiply both sides of this equation by -9 divided by 40, and that is going to leave us with X2 equal to -3 divided by 10. There is a bit of an issue here because the moment we take the square root of both sides of the equation, we will go ahead and end up with an imaginary number. Therefore, there is not going to be any inflection points, but we can still check the concavity around the 0.0. The reason why we can do this is because we still have an X term within a denominator, and the restriction of this derivative is 0. So we have to check the concavity around this restriction. So, we will take a testing point in the neighborhood of 0, plug it into the second derivative, and depending on the value, we will determine if the graph is going to be concave down or concave up. A testing point we can take to the left of 0 is -1. F of -1 is going to give us the value of 52 divided by 9. Since this is positive, the graph will be concave up within this region. And if we take the testing 0.1 and plug it into the second derivative, F1 will give us the value of 52 divided by 9. This also tells us that the graph is going to be concave up within this region. So, let's go ahead and just graph the concavity with the regions of increase and decrease. Now, we know the graph is concave up between 0 and the minimum value square root 3 divided by 2. And it is decreasing to this local minimum. Then we know that the graph will be increasing to positive infinity afterward. And if we mirror this behavior because of the symmetry about the y axis, this is going to be the entirety of the given of the given functions graph. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Use the guidelines of this section to make a complete graph of f.
f(x) = 2 - 2x^2/3 + x^4/3

Key Concepts

Graphing Functions

Polynomial Functions

Critical Points and Derivatives

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