Finding Critical Points

In E...

Question

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

g(x) = √(2x − x²)

Accepted Answer

Hello. In this video, we want to determine all the critical points and all the domain end points for the given function H X equal to the square root of 3 X minus X2. Then we want to choose which of the following accurately represents this information. So, let's go ahead and first start this problem by determining the end the end points of the domain. So we are given H X equal to the square root of 3X minus X2. Now, since we are given a square root function, we want to make sure that the inside of the function is not negative. This means that the insight argument 3X minus X squared must be strictly greater than or equal to 0. So what we are going to do now is we are going to solve for the values of X that cause this inequality to be true. The first thing we can do is factor out an X from the left hand side of the inequality, leaving us with X multiplied by 3 minus X, and this is greater than or equal to 0. Now, the smallest value of X that causes this function to be true is going to be X is equal to 0. When we plug in 0, we get 0 on the left hand side of the equality, and that is going to be true. And the greatest value of X that causes this inequality to be true is X is equal to 3. This will give us 0 on the left hand side of the inequality, but if we plug in anything that is greater than 3, we get a negative number, which causes the inequality to be true. So since, or to be false, so since the smallest value is X is equal to 0, and the greatest value is X is equal to 3, the domain endpoints are going to be from 0 to 3. Now what we want to do is we want to determine the critical points of this function. Now, in order to determine the critical points, we want to find the values of X that cause the derivative to be zero. So we will need to take the derivative of this function. Before we do that, let's rewrite the function in an exponential form. That is going to give us H of X is equal to the quantity, 3 X minus X2d raise to the half power. Now we will take the derivative with respect to X. By using the chain rule, we get H of X is equal to 1/2 multiplied by 3 X minus X squared, raising the power of negative half, multiplied by the derivative of the innermost function, which is 3 minus 2 X. By combining these terms together, we can simplify the function to be 3 minus 2 X divided by 2 multiplied by the square root of 3 X minus X squared. Now, we can obtain the critical points by finding the values of X that cause the numerator to be 0. That will give us 3 minus 2 X equal to 0, and by solving for X, we get X's equal to 3 halves. Now, 3 halves has a decimal value of 1.5, and that is within our domain endpoints. So X is equal to 3/2 is going to be a critical point of the function. Another thing we have to consider are the values of X that cause the denominator to be zero. Division by 0 gives us undefined values, so let's go ahead and solve for those values of X. We will take our denominator 2 multiplied by the square root of 3X minus X2, set it equal to 0, and solve. Divided by 2 leaves us with the square root of 3 X minus X2 equal to 0. If we square both sides of this equation, we get 3 X minus X2 equal to 0, and by factoring out an X, we get X multiplied by 3 minus X equal to 0. Now what we can do is we can set each factor of X equal to 0, and that is going to give us X is equal to 0 and 3 minus X is equal to 0. X is equals 0 will be a second critical point within our domain endpoints, and by solving the right-hand equation, we get X's equal to 3, which is also within our domain endpoints, since 3 is included. So in total we have 3 critical points. X is equal to 3 halves, 0 and 3, and our domain endpoints are going to be from 0 to 3. With that being said, the solution to this problem is going to be C. So I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

g(x) = √(2x − x²)

Key Concepts

Critical Points

Domain of a Function

Derivative of a Function

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