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) = 1 / x + ln x

Accepted Answer

Welcome back everyone. Determine the critical points of the function F of X equals 2 LNX minus X2. We're given four answer choices A says 1.1, B 11, C 0.0, and 11 D 0.0, and 1.1. Whenever we want to identify the critical points of a function, first of all, we have to find its derivatives. So we are differentiating F of X. Let's identify the derivative of 2 L N X minus X2. So now we are going to differentiate the natural log. The derivative of that is 1 divided by X. So we get 2 divided by X, and the derivative of X squared is 2 X. So we get 2 divided by X minus 2 X. And from here we have two cases. The first one is when does the derivative does not exist in the domain of the function. Notice that our domain. Is X greater than 0 because we have a natural log function, so we want to make sure that X is positive, right? And now our derivative doesn't exist when X is 0 and that's not in the domain, right? So since it's not in the domain of the original function, we don't need to check this condition. The second condition is to check when the derivative is equal to 0, and that's what we have to do. So we are going to set it equal to 0. If F of X if F of X is equal to 0, then this means that 2 divided by X minus 2 X is equal to 0, and we're going to find the common denominator. We're going to get 2 minus 2 x squared divided by X is equal to 0. And in this case we have a rational expression to solve, right? So we're setting the numerator equal to 0, the denominator is not equal to 0. We can simply say that 21 minus X squared is equal to 0. So we're using a factored form, right? And now we can use a difference of squares factorization, so we get 21 minus X multiplied by 1 + X is equal to 0. Now this equation has two solutions. Either X is positive or -1. But once again, let's recall that x must be positive, so we only have one solution, and that solution is X equals 1. So we have identified our critical point. We just want to identify its Y coordinate, and the Y coordinate is going to be found by evaluating F at 0.1. So we get 2 LN of 1 minus 1 squared. LN of 1, well, this gives us a value of 0. And then we are subtracting one so we get -1. So the point of interest. Or basically the critical point is 1.1, and our final answer to this problem corresponds to the answer choice B. Thank you for watching.

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

ƒ(x) = 1 / x + ln x

Key Concepts

Critical Points

Derivative

Natural Logarithm

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