Finding max/min of 3d graphs with chain rule

In summary, the conversation is about problem number 2 in a practice midterm, where the person is seeking help. The problem involves combining max and min with multivariables and the chain rule, which the person has not done before. Another person suggests using t as the variable and differentiating with respect to t to find the two critical points, one of which is a minimum and the other a maximum. The conversation also briefly mentions the difficulty with the professor's handwriting.
  • #1
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http://math.berkeley.edu/~theojf/Midterm2Practice.pdf

can someone please help me on problem number 2 of the link above?
apologies for the bad handwriting. my professor is just horrible with that.

i've done max and min with multivariables before and I've done chain rule , but I've never combined the two. also, in that problem, the only critical point seems to be a saddle point, so I'm not sure what the "lowest" point is referring to
 
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  • #2
You don't need to use the chain rule- in fact you don't need to worry about multi-variables at all. Replace x and y in z(x,y) with their expressions in terms of t and differentiate with respect to t. There are two critical points. One is a minimum and the other a maximum.
 

What is the chain rule and how does it apply to finding maximum and minimum points on a 3D graph?

The chain rule is a mathematical concept that allows us to find the rate of change of a function with respect to another function. In the context of finding max/min points on a 3D graph, the chain rule helps us to determine the direction in which the function is increasing or decreasing at a given point.

What are the steps involved in using the chain rule to find maximum and minimum points on a 3D graph?

The steps involved in using the chain rule to find max/min points on a 3D graph are as follows:

  • First, identify the function that represents the 3D graph.
  • Next, find the partial derivatives of the function with respect to each variable.
  • Then, set each partial derivative equal to zero and solve for the variables.
  • Finally, plug in the values for the variables into the original function to find the coordinates of the max/min point.

What are some common mistakes to avoid when using the chain rule to find maximum and minimum points on a 3D graph?

Some common mistakes to avoid when using the chain rule to find max/min points on a 3D graph include:

  • Mistaking the function for its partial derivatives.
  • Not setting the partial derivatives equal to zero before solving for the variables.
  • Forgetting to check the second derivatives to confirm if the point is a max or min.
  • Using the incorrect order of variables when finding the partial derivatives.

Can the chain rule be used to find maximum and minimum points on any type of 3D graph?

Yes, the chain rule can be used to find max/min points on any type of 3D graph as long as the function representing the graph is differentiable and the partial derivatives can be found.

Are there any other methods for finding maximum and minimum points on 3D graphs other than using the chain rule?

Yes, there are other methods such as gradient descent and Lagrange multipliers that can also be used to find max/min points on 3D graphs. However, the chain rule is a commonly used and efficient method for this purpose.

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