Let f(x) = 4√x - x.
Find all points on the graph of f at...

Question

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line is horizontal.

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with derivatives. This problem says for the function H of X, which is equal to 5 multiplied by the square root of X, minus 3 X, identify all points in the graph of H where the graph has a horizontal tangent. We give 4 possible choices as our answers. For choice A, we have the point of -25 divided by 36, 25 divided by 12. For choice B, we have the point of -25 divided by 36, 25 divided by 12. For choice C, we have the point of 25 divided by 36, minus 25 divided by 12, and for choice D we have the point of 25 divided by 36, 25 divided by 12. Now we're asked to identify all points on the graph of H, where the graph has a horizontal tangent. So in order to have a horizontal tangent, that means our tangent line has to have a slope of 0. In order to determine the slope of our tangent line at a given point, we need to take a derivative of our function H of X. So we're going to start off by looking for H of X, and this is going to be equal to the derivative with respect to X of the quantity of 5. Multiplied by X raised to the 12. So here I've just rewritten the square root of X as X raised to the 1/2 power, then we'll have minus 3 multiplied by X. So, I'm gonna have to take a derivative of each of these terms that this is going to be equal to. For the first term, we're going to use our constant multiple and power rule. So, we're going to have 5, multiplied by the X opponent, which is 1/2, and that gets multiplied by X raised to the 1/2 minus 1 power, which means that we have X raised to the minus 1/2 power. Then we have minus the derivative of 3 multiplied by X, and when I take the derivative of X with respect to X, I get 1, so I'll have 3 multiplied by 1, so this is just 3. So we can simplify this expression, so this is going to be equal to 5, divided by the quantity of 2 multiplied by the square root of X in quantity minus 3. So here we've just rewritten X-rays to the minus 1 half power as 1 divided by the square root of X. Now, in order to find the places where the slope of our tangent line is zero, meaning we have a horizontal tangent line, we need to set this derivative equal to 0 and find all possible values for X. So, in order to solve this for X, we'll first add 3 to both sides of this equation. So we'll have 5 divided by the quantity of 2 multiplied by the square root of X in quantity, and that is going to be equal to 3. Now we want to get rid of the fractions, we'll multiply both sides by 2 multiplied by the square root of X. And in doing so, we will get 5 is equal to 6 multiplied by the square root of X. To isolate X, we'll divide both sides by 6. And so we'll have 5 divided by 6, is equal to the square root of X. And so now we just need to square both sides to solve for X. And in doing so, we'll get X is equal to. 25 divided by 36. So that is the X coordinate of our point where uh where we have a horizontal tangent line. So now that we have the X coordinate, we just need to find the Y coordinate, and so that means we're just going to substitute N for X, our value of 25 divided by 36, into our function H of X. So, we're gonna be looking at each of 25, divided by 36. And this is going to be equal to 5, multiplied by the square root of the quantity of 25. Divided by 36 in quantity, minus 3 multiplied by 25 divided by 36. We just need to simplify this expression. So this is going to be equal to the square root of 25 divided by 36 is 5 divided by 6, and when we multiply that by 5, we'll get 25 divided by 6 for the first term, then we will have minus 25. Divided by 12, since 36 divided by 3 will give me 12 in the denominator. And when I subtract these two fractions, this is going to be equal to 25 divided by 12. So that is the Y coordinate for our point that we're looking for. And now we can write this in point notation. So this is going to be the point of 25 divided by 36, 25 divided by 12. And so that is the answer that we are looking for. And when we compare our answer here to our answer choices, we see that the correct answer choice is answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Let f(x) = 4√x - x.
Find all points on the graph of f at which the tangent line is horizontal.

Key Concepts

Derivative

Critical Points

Tangent Line

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