F(x) = -(1)/(4)x^4 - x^3 + 5x^2 + 24x - 5 Determine whether...

f(x) = -\frac{1}{4}x^4 - x^3 + 5x^2 + 24x - 5 Determine whether f(3) is a local maximum or a local minimum. (Use x = 2.9 and x = 3.1 in your calculations.) Local minimum Local maximum

Transcript

00:01 Okay, so in this exercise we have our function f of x, y equal to 3x plus y, defined on the region x squared plus y squared less than or equal to 2.

00:15 Now, we want to find the maximum and the minimum of f on this region.

00:22 Well, clearly the maximum and the minimum are going to be achieved along the boundary.

00:29 Of this region here, and the boundary is just a circle.

00:35 The circle with center the origin and radius square root of 2.

00:43 Perfect.

00:44 Okay, at this point we can use lagrange multipliers to find the maximum and the minimum.

00:51 Well, to do this, let me define our constraint function g of x, y equal to x squared plus y squared minus 2.

01:02 Perfect.

01:03 Now, to use lagrange multipliers we need to solve the system of equations gradient of f equal to lambda gradient of g and g identically equal to 0.

01:19 Now, the gradient of f is 2 comma, okay, f is 3x plus y, so 3 comma 1.

01:30 Perfect.

01:31 The gradient of g, well, this one is 2x comma 2y.

01:39 Perfect.

01:40 So, our system of equations is, okay, 3 equal to 2 lambda x, 1 equal to 2 lambda y, and x squared plus y squared minus 2 equal to 0.

02:02 Let me call this equation 1, this one 2, this one 3.

02:06 Okay, now, let's consider the quotient of the first equation and the second one.

02:17 Well, this one gives us a new equation which is 3 equal to x over y...

Question

f(x) = -\frac{1}{4}x^4 - x^3 + 5x^2 + 24x - 5 Determine whether f(3) is a local maximum or a local minimum. (Use x = 2.9 and x = 3.1 in your calculations.) Local minimum Local maximum

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