Problem 2 (20pt). Let f(x, y) = 2x - 4xy + y. a. 10pt Find...

Problem 2 (20pt). Let $f(x, y) = 2x - 4xy + y$. a. 10pt Find all local extrema (minimum, maximum, and saddle points) for $f$ on $R^2$. b. 10pt Find the global maximum and global minimum of $f$ on the region bounded by $x$-axis, $y$-axis, and the line $x + y = 1$.

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Transcript

00:01 In this question we are given a region, given a function f of xy is equal to 2x square minus 4x plus y square minus 4y plus 1.

00:16 Bounded by the region having points, x is equal to 0, y is equal to 2 and y is equal to 2x.

00:31 Now, for the first subpart, we need to obtain all the critical points of this function.

00:41 For that, let us first differentiate this function, f -dash -of -xy, that is, differentiated with respect to x and with respect to y.

00:54 Now, differentiating with respect to x, we get curly f over curly x is equal to curly over curly x of 2x2x2 ,000, minus 4.

01:06 4x plus y square minus 4y plus 1.

01:12 Inferentating we get 4x minus 4.

01:20 Also the derivative with respect to y is given by curly over curly y of 2x2x2 minus 4x plus y2x plus y2.

01:34 Solving this we get 2y minus 4.

01:39 Now at the critical points the derivatives are used.

01:44 Equal to 0, which implies curly f over curly x is equal to 0, that is 4x minus 4 is equal to 0.

01:57 This gives the value of x as 1.

02:01 Also, the derivative with respect to y is equal to 0, and we get to y minus 4 is equal to 0.

02:09 Therefore, y is equal to 2.

02:11 Now, therefore the critical point is 1, 2.

02:16 Another critical point was 0 .0.

02:22 Now according to the given lines, the third critical point is 1 .0 .2, according to the values of x...

Question

Problem 2 (20pt). Let $f(x, y) = 2x - 4xy + y$. a. 10pt Find all local extrema (minimum, maximum, and saddle points) for $f$ on $R^2$. b. 10pt Find the global maximum and global minimum of $f$ on the region bounded by $x$-axis, $y$-axis, and the line $x + y = 1$.

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