Finding Critical Points

In E...

Question

Finding Critical Points

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

y = x² + 2/x

Accepted Answer

Welcome back, everyone. Find all critical points and domain end points for the function H of X equals 2 X to the power of 5 minus 5 divided by X. First of all, let's focus on the domain of this function. Let's recall that division by 0 is undefined. We have the irrational term 5 divided by X. So we can see that our denominator cannot be equal to 0, right? Because division by 0 is undefined. So the domain of this function is all real numbers except from 0. Meaning x belongs to the interval from negative infinity up to 0. And from 0 up to infinity. Let's label this expression as our first answer to this problem. For the second part of the problem, we want to identify the critical points, and this means that we have to differentiate H of X. That's the derivative of 2 multiplied by X to the power of 5. Minus the derivative of 5 divided by X. We can rewrite this as 2 multiplied by the derivative of X to the power of 5 minus 5 multiplied by the derivative of X to the power of -1, using the reciprocal rule. Applying the power rule, we get 2 multiplied by 5 X to the power of 4 minus 5 multiplied by -1. Multiplied by x to the power of -2. So we simply bringing down the exponent and decreasing the original exponent by one unit. This gives us 10 X to the power of 4. Plus 5 X to the power of -2. Which can be written as 10x the power of 4 plus 5 divided by x squared. And now finding the common denominator we get. 10 x to the power of 6. Plus 5 divided by X2. Now to get the critical points, we have to recall that either H of X. Is equal to 0, we set it to 0. Or of X, so the same derivative as H X equals 0 is now. Set to does not exist, or basically we can say that this derivative does not exist. Now, the derivative does not exist when X is 0, right? Because the division by zero is undefined, but 0 is not in the domain. So we don't need to check the second condition and we only focus on each prime of X is equals to 0. Which means that we want to set our numerator equals to 0, so send x to the power of 6 + 5. Is equal to 0, right? If we want to solve this equation, well, we are going to subtract 5 from both sides. Which gives us 10x the power of 6 equals -5, and then dividing both sides by 10, we get X to the power of 6 equals -5 divided by 10 or -12. Now we have X-rays to an even exponent, and this gives us a negative number. So this equation has no solutions. Because raising a real number to an even exponent must give us a positive number and not a negative number, meaning this equation has no solutions and therefore we have no. Critical points Which is the second answer to this problem. That's it. We have solved the problem, and thank you for watching.

Finding Critical Points

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

y = x² + 2/x

Key Concepts

Critical Points

Derivative

Domain of a Function

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