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 has slope -1/2.

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to consider the function G of X, which is equal to 5 multiplied by the square root of X minus 2 X, and to determine the points on the graph of G, where the slope of the tangent line is equal to -1. We're given 4 possible choices as our answers. For choice A, we have the point of 0.25 divided by 4. For choice B, we have the point of -25 divided by 4.0. For choice C, we have the point of 25 divided by 4.0. And for choice D, we have the point of 0.25 divided by 4. Now, we're asked to determine the points on the graph of G, where the slope of the tangent line is equal to -1. And in order to determine the slope of the tangent line at a given point, we need to take a derivative of our function G of X. So we're going to start off by calculating G of X. And that is going to be equal to the derivative with respect to X of the quantity of 5 multiplied by X raised to the 1/2 power. Minus 2 X. And here we just rewritten the square root of X as X-rays to the one half power. So now I'm going to need to apply the derivative to each of these terms. So doing so, this is going to be equal to. For the first term, we're going to need to use our constant multiple and power rule for derivatives. So recall for that, we're going to have our constant, which is 5, multiplied by the X opponent, which is 12, and that gets multiplied by X raised to the 1/2 minus 1 power, so we'll have X raised to the minus 12 power. Then we'll have minus, and we'll have to take the derivative of the second term, which is the derivative with respect to X of 2 X. So we'll have 2 multiplied by the derivative of X with respect to X, which is 1, so we'll have 2 multiplied by 1, so we'll just get 2 as that second term. So we can simplify this expression, and this is going to be equal 2. 5 divided by the quantity of 2 multiplied by the square root of X in quantity minus 2. And here I've just rewritten X-rays to the minus 15 power as 1 divided by the square root of X. Now we need to determine the points on the graph where the tangent line, the slope of the tangent line is equal to -1. So that means I need to take our derivative here and set it equal to -1 and solve it for X. So doing so, we'll start by adding 2 to both sides of this equation. And so that means I'm going to have 5 divided by the quantity of 2 multiplied by the square root of X is equal to 1. Now, to get rid of the fraction, I'm going to multiply both sides of the equation by 2 multiplied by the square root of X. And so I'll have 5 is equal to 2 multiplied by the square root of X. I'll then need to divide both sides by 2 to isolate X, so we'll have 5 divided by 2 is equal to square root of X. And then sulfur X will just square both sides of the equation. In doing so, I'll get X is equal to 25 divided by 4. And so that is the X coordinate of the point that I'm looking for. And now I need to determine the Y coordinate for that point. And to do that, we'll just substitute in 25 divided by 4 for X into our function of G of X. So we're going to be calculating G of 25 divided by 4. And that is going to be equal to 5, multiplied by the square root of the quantity of 25 divided by 4 in quantity, minus 2 multiplied by 25 divided by 4. So, we can simplify this expression, so this is going to be equal to. For the first term, the square root of 25 divided by 4 reduces to 5 divided by 2. I want to multiply that by 5, I'll get 25. Divided by 2. Then we'll have minus for the next term, we will have 25 divided by 2, since we can um Divide 4 by 2 to get 2 in the denominator. And when I subtract these two fractions, this is going to be equal to 0. And so that that is the Y coordinate for our point that we're interested in. So now we can just rewrite our final answer as a point, so we're gonna have the point of 25. Divided by 4.0. So that is the final answer to this problem. And when we compare our answer here to our answer choices, the correct answer choice corresponds to 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 has slope -1/2.

Key Concepts

Derivative

Tangent Line

Finding Critical Points

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