Locating critical points Find the critical points of the follo...

Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x² √(x + 5)

Accepted Answer

Welcome back, everyone. Determine the critical points of the function F of X equals 2 X2 square root of 2X minus 4. We're given 4 answer choices A 2.0, B 0.0, and 2.0, C 4.64, and D, there are no critical opens. The first step is to define the domain of the original function. So as we can see it as a product of two functions and we're interested in the radical part, we have to recall that square root of a number essentially must have a value that is nonnegative under the radical, right, so 2 x minus 4 must be greater than or equal to 0, which essentially means that. 2 x minus 2 is greater than or equal to 0, and this means that the value X minus 2 must be greater than or equal to 0. Or simply speaking, X must be greater than or equal to 2, so that's the domain. Our next step is to differentiate the function, and we can see that using the product rule. So we're taking the derivative of the first function, so 2 x squad, multiplying by the second function, which is square root of 2 x minus 4, and we are adding the first function which is 2x2 multiplied by the derivative of the second function. So now we're going to rewrite 2 x minus 4. Square root of 2 x minus 42 x minus 4 raises the power of 1/2, right, because we want to apply the power rule. And now let's go ahead and solve. So the derivative of X where 2 Xs, we're going to get 4 X for the first derivative. Which is multiplied by the radical term 2 x minus 4 square root of that. La 2 x 2, which is multiplied by. Now let's apply the power rule, right? We're going to bring down the exponent 12. Let's decrease the original exponent by 1 unit according to the power rule. So that's going to be 1/2, and now the inner function is a composite function to x minus 4, right, is within the radical. So we have to multiply by the derivative of 2 x minus 4 according to the chain rule. And now let's go ahead and valuate the result. So now the derivative of 2x minus 4 is 2 according to the power rule, and the derivative of a constant is 0. So 2 divided by 2 gives us 1 so we can cross those out, and this means that the overall derivative is 4 x square root of 2 x minus 4 plus. 2 X squad and then we're going to divide that by square root of 2 X minus 4 because we have a negative exponent, we can write it in the denominator and 1/2 corresponds to square root. Now, if we find the common denominator, that's going to be square root of 2 x minus 4, and then we're going to have 4 X C 2 x minus 4 plus 2 X squad in the numerator, right? How did we get it? Well, essentially we multiplied and divided the first term by square root of 2 x minus 4, so we had to square the radical term. And now let's distribute, so 8 X2 + 2 x 2 is 10 x 2, and then we are subtracting 16 X. And then we are dividing by the common denominator of 2 X minus square root of 2 X minus 4. So we have two cases. Number one, we set it equal to 0 to find the critical points, which means that 10 x minus 1610 X squad actually, right? So let's include the square. So 10 x 2 minus 16 X is equal to 0. We can factor it out as 2 X multiplied by 5 X minus 8 is equal to 0. And this gives us two solutions because we have two factors. Either X is equal to 0, the first factor, right, or X is equal to 8 divided by 5, right? So we are going to get X equals 85. Now, notice that thinking about these values while they are outside of the domain, right? So they are not. Critical points. Because they are both less than 2, right? So, because they're both less than 2, this means that they are not critical points. Now, the second condition needs to be checked. We want to see where the derivative does not exist. And this means that, well, since we have a rational function, we want to set our denominator equal to 0, which means that the term under the radical must be equal to 0. So 2 x minus 4 equals 02 X equals 4, and X equals 2, right? So we have got our first vertical point. It is within the domain, and we simply want to identify F at 0.2 to identify its Y coordinate, which gives us 2 multiplied by 2 squared multiplied by a square root of 2 multiplied by 2 minus 4, which is 0, right, because we're multiplying by 0. And this means that our critical point has X of 2 and Y of 0. So the only critical point is 20, which corresponds to the answer choice A. Thank you for watching.

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x² √(x + 5)

Key Concepts

Critical Points

Derivative

Product Rule

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