To find the extreme values of a function f(x, y) on a curve...

To find the extreme values of a function $f(x, y)$ on a curve $x=x(t), y=y(t),$ we treat $f$ as a function of the single variable $t$ and use the Chain Rule to find where $d f / d t$ is zero. As in any other single-variable case, the extreme values of $f$ are then found among the values at the a. critical points (points where $d f / d t$ is zero or fails to exist), and b. endpoints of the parameter domain. Find the absolute maximum and minimum values of the following functions on the given curves. Function: $f(x, y)=x y$ Curves: i. The line $x=2 t, \quad y=t+1$ ii. The line segment $x=2 t, \quad y=t+1, \quad-1 \leq t \leq 0$ iii. The line segment $x=2 t, \quad y=t+1, \quad 0 \leq t \leq 1$

To find the extreme values of a function $f(x, y)$ on a curve $x=x(t), y=y(t),$ we treat $f$ as a function of the single variable $t$ and use the Chain Rule to find where $d f / d t$ is zero. As in any other single-variable case, the extreme values of $f$ are then found among the values at the
a. critical points (points where $d f / d t$ is zero or fails to exist), and
b. endpoints of the parameter domain.
Find the absolute maximum and minimum values of the following functions on the given curves.
Function: $f(x, y)=x y$
Curves:
i. The line $x=2 t, \quad y=t+1$
ii. The line segment $x=2 t, \quad y=t+1, \quad-1 \leq t \leq 0$
iii. The line segment $x=2 t, \quad y=t+1, \quad 0 \leq t \leq 1$

Key Concepts

Absolute Extreme Values

The absolute extreme values of a function on a curve are the highest and lowest function values obtained either at critical points or at the endpoints of the parameter domain. Finding these extremes involves computing and comparing the function values at all candidate points, a process that ensures all possible extrema are considered within the given limits.

Critical Points

Critical points are the values of the parameter t for which the derivative of the function f(x(t), y(t)) with respect to t is zero or does not exist. These points are the candidates for local extreme values. In optimization problems along curves, determining where the derivative equals zero or fails to exist is essential for further testing and finding extreme values.

Endpoints of Parameter Domain

Endpoints refer to the boundaries of the interval for the parameter t over which the curve is defined. In single-variable optimization, extreme values can occur at these endpoints. Thus, evaluating the function at the endpoints of the parameter domain is as important as checking critical points when seeking absolute maximum or minimum values.

Chain Rule

The Chain Rule is a fundamental tool in calculus that allows us to differentiate composite functions. When dealing with a parameterized function, it is used to calculate the derivative of f(x(t), y(t)) with respect to t by multiplying the partial derivatives of f with the derivatives of x and y with respect to t. This step is crucial for identifying where the function’s rate of change is zero or undefined, which may correspond to maximum or minimum points.

Parameterization of Functions

This concept involves expressing a multivariable function in terms of a single variable by substituting in the parameterized forms of the original variables. When a curve is represented by x = x(t) and y = y(t), the function f(x, y) becomes f(x(t), y(t)). This transformation simplifies the problem to single-variable calculus, facilitating the analysis of extreme values along the curve.

Transcript

00:01 Now we want to minimize the function x times y over a path.

00:07 So the path is x equals 2 t and y equals t plus 1.

00:12 And then the first line is unbounded and then we have different segments of the line.

00:20 So when we look at the unbounded case, so the ft t is y times the x d t plus x times the y to t.

00:29 The x d t is 2, the y d t is 1.

00:32 So that means the fd t is 2y plus x or let's see this is 4 t plus 2 plugging in values for for these.

00:45 That says that we have critical points at a critical point at t equals minus 1 half, which means that x is minus 1 and y is 1 half.

00:56 And that says that that critical point our function is minus one half.

01:03 Now that's our critical point but now we can look at our domain and so as t goes to plus or minus infinity we can see that x goes and y go both go to plus or minus infinity and then so f goes to infinity because we're multiplying it.

01:22 So if t goes to plus infinity then of course these go to plus infinities if t goes to minus infinity then these go to minus infinity then these go to minus infinities, but we're multiplying them.

01:33 So the maximum value here is infinite.

01:37 But the minimum is not minus infinity because it's kind of a, you know, hyperbola type shape.

01:45 So the minimum value is actually right here, at our critical point of minus 1 1 half.

01:51 Now, the first problem, we say that, ok, let's say we have a line segment minus 1 to 0...

Please give Ace some feedback