Video Transcriptfor this problem, we are asked to find the minimum of the function F. Of X. Y equals X times Why is subject to the constraint that X squared plus Y squared equals nine. So what we want to do is we have that our G of X. Y. Our constraint function can be expressed as X squared plus Y squared minus nine. And we want to then find lambda X. And Y. Such that the gradient of F equals lambda times the gradient of G. So taking the derivatives. That means that we need Why to equal? Let's see here. That would be we want Y to equal two times lambda X to lambda X. Here. And then with respect to why we'll get that X must be equal to lambda. Why? Which in turn, we know that why equals 2 λ X. So you can plug that in. We'd have that X equals And is misbehaving X would equal two times λ Times 2 λ X. So we have that. We can actually write this as being first of all, that one is going to equal four times lambda squared. So λ is going to equal plus or -2 or actually plus or minus 1/2 Because we divide through both sides by four square root. All right. So, we have that then. And we also have that Y equals two times lambda times X. Now we need to figure out what X is. We can do that. Bye. We know that. What why is in terms of X. And we know what lambda is. So we can plug that into our expert. Plus Y squared equals nine equation there. So we get that X squared plus four lambda squared X squared must equal nine for lambda square two, X squared is going to just go to X squared because we know that land is plus or minus one half. So we get that two X squared equals nine or in turn that X squared equals nine over to. So X equals plus or minus route or plus or minus three over root two, Say 1 2nd here. All right. Yeah. So we have X equals plus or minus three over root two, which then means that why is going to equal plus or minus? Um two times 1/2. So yeah, it's going to be equal to then plus or minus. Um Actually, it's just going to shake out to being plus or minus X. So plus or minus three over root two as well. So what we need to do then is evaluate our function at each one of those different possible points. Or we could just recognize that since we have we are dealing with F of X. Y. That to get the maximum we'll plug in the largest values. So we would have f of three over root two Times three Over Root two. So that's going to be nine over root two times two. So 9/2 is the maximum and then the minimum evaluated at negative three over root two, Positive three over route to being careful because if we multiply the two negatives together, we would end up with a positive value again. But if we take a negative and a positive then we'll get negative 9/2. And that also equals the value of the function at the 0.3 over root two times -3 Over Root two. And I'll note that that value was positive three over route to positive three over root two. It's going to be the same thing as the value of the function at negative three over root two, negative 3/2. Show Knowledge Booster Learn more about Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below. Recommended textbooks for you  Calculus: Early Transcendentals ISBN:9781285741550 Author:James Stewart Publisher:Cengage Learning Thomas' Calculus (14th Edition) ISBN:9780134438986 Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir Publisher:PEARSON Calculus: Early Transcendentals (3rd Edition) ISBN:9780134763644 Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz Publisher:PEARSON Calculus: Early Transcendentals ISBN:9781319050740 Author:Jon Rogawski, Colin Adams, Robert Franzosa Publisher:W. H. Freeman Precalculus ISBN:9780135189405 Author:Michael Sullivan Publisher:PEARSON Calculus: Early Transcendental Functions ISBN:9781337552516 Author:Ron Larson, Bruce H. Edwards Publisher:Cengage Learning Calculus: Early Transcendentals ISBN:9781285741550 Author:James Stewart Publisher:Cengage Learning Thomas' Calculus (14th Edition) ISBN:9780134438986 Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir Publisher:PEARSON Calculus: Early Transcendentals (3rd Edition) ISBN:9780134763644 Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz Publisher:PEARSON Calculus: Early Transcendentals ISBN:9781319050740 Author:Jon Rogawski, Colin Adams, Robert Franzosa Publisher:W. H. Freeman Precalculus ISBN:9780135189405 Author:Michael Sullivan Publisher:PEARSON Calculus: Early Transcendental Functions ISBN:9781337552516 Author:Ron Larson, Bruce H. Edwards Publisher:Cengage Learning How do you find the absolute minimum and maximum of a function?Finding the Absolute Extrema. Find all critical numbers of f within the interval [a, b]. ... . Plug in each critical number from step 1 into the function f(x).. Plug in the endpoints, a and b, into the function f(x).. The largest value is the absolute maximum, and the smallest value is the absolute minimum.. What functions have an absolute maximum or minimum?A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.
How do you find the maximum and minimum value of a function in a range?We have to find the maximum or minimum value of the function. The maximum or minimum of a quadratic function occurs at x = -b/2a. If a is negative, the maximum value of the function is f(-b/2a). If a is positive, the minimum value of the function is f(-b/2a).
What is the maximum minimum value of the function?Here, the maximum value f(x) at x = 1 is called the absolute maximum value, global maximum or greatest value of the function f on the closed interval [0, 1]. Similarly, the minimum value of f(x) at x = 0 is called the absolute minimum value, global minimum or least value of the function f on the closed interval [0, 1].
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Find the absolute maximum and minimum values of the function subject to the given constraints
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