Does ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x² have any inflection points...

Question

Does ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x² have any inflection points? If so, identify them.

Accepted Answer

Welcome back, everyone. Find the inflection point or points of F of X equals 7 divided by 30 X to the power of 6 minus 35 divided by 12 x to the power of 4 minus 126 x 2. If we want to identify the inflection points, well, we have to find the second derivative of the function to begin with. So let's begin with the first derivative. We're going to apply the power rules. So first we'll, let's factor out 7 divided by 30, and now the derivative of X, the power of 6 is 6 X to the power of 5, according to the power rule. Then we are subtracting our next constant 35. Divided by 12, which is multiplied by. The derivative of X is the power of 4, and according to the power rule, we get 4 X cubed. Then we are subtracting 126, the derivative of X quad is simply 2 X. Now from here, we are going to identify our second derivative. F of X, so we get 7 divided by 30, multiplied by 6 multiplied by the derivative of X to the power of 5 is 5 X to the power of 4. Then we have minus 35 divided by 12 multiplied by 4 multiplied by the derivative of X cubed is VX2. And then we have minus 126 multiplied by 2 because the derivative of X is simply 1. And now let's simplify. So the second derivative is going to be equal to 7 X to the power of 4 minus 35 x 2 minus 252. So now what we want to do is simply set it equal to 0 because we know that the inflection points occur where the second derivative is 0 or does not exist. It is a polynomial, so we're only checking when it is equal to 0. And essentially we're going to use the quadratic equation instead of the 4th. Exponent, right? And what we can do is simply say that C2d is equal to. Or actually Z is equal to. X2. So we're going to use substitution. This means that F of C can be expressed as 7 z2, right, because when we square C we get X the power of 4 minus 35 C minus 252. And now we can also factor it out. So we're going to get 7 in C C squad minus 5C. Minus 36. And now we can also factor it out further. So we're going to get 7 in C. + 4. Multiplied by z minus 9. So when we set it equal to 0 in this factored form, we can see that we have two solutions either C1 is equal to -4, or we have another solution Z2 is equal to 9. But let's recall that C is X2, so it's always greater than or equal to 0, meaning Z1 is not our solution because it's negative. So we're only considering Z2. And this means that X squared is equal to 9. And if we take square roots of both sides, this gives us X equals or -3. So we have two candidates for a point of inflection. What we want to do is simply identify if these are indeed points of inflections, we're going to take the second derivative. F of X. Which we can express as 7. Multiplied by C +4, which is X2 + 4, multiplied by X2 minus 9. And we want to check what signs we get in appropriate intervals because we have two candidates -3. And 3, this gives us 3 intervals to consider on the number line. So, first of all, we want to consider an interval between negative infinity and -3. So we can take any value that we want. We can take -4. So let's suppose that x equals -4. This means that our first factor is positive. It's always positive, right, because we have X2 + 4 and the minimum value of X2 is 0, so we can see that the first factor is always positive in every interval. And now what we want to do is simply check what's the sign of the second factor. So if we take X equals -4, when we squared, we get 16 and 16 minus 9 is positive. This means that we have two positive factors between negative infinity and -3, and the product of two positive factors is going to be positive. This means that the function is concave up. Now, between -3 and 3, we can take 00 minus 9 gives us a negative, second factor. And specifically negative multiplied by positive gives us a negative value, meaning the function is concave down because the second derivative has a negative sign, and then we want to take any value that is greater than 3, such as 4. So 42 minus 9 is positive and this means that our X2 minus 9 is positive, giving us a positive product. So. The function is concave up again from 3 up to infinity. As we can see, it goes from up to down to up meaning. Since the second derivative changes at sign, both of these are. Points of inflection. And now what we want to do is simply identify the corresponding Y coordinates using the original function F. So we want to evaluate F at. -3, which is 7 divided by 30, multiplied by -3 to the power of 6. -35 divided by 12 multiplied by -3 to the power of 4 minus 126 multiplied by -3 squad. Now let's evaluate the result. We are going to get -1,200.15. And now if we evaluate. F At a.3, that is going to be the same value. How do we know that? -1,200.15. Well, as we can see, it is an even function. Basically this means that F of X is equal to f of negative X, right? We have X the power of 6, X the power of 4, and X squad. If we take negative X for each, we're going to get the same function F of X. So we don't need to repeat all those calculations. We can just immediately say that. The Y coordinate will be the same at X equals 3. So, our inflection points are simply -3, 1,200.15, and Positive 3, 1,200115. So those would be our final answers to this problem. Thank you for watching.

Does ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x² have any inflection points? If so, identify them.

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