Unit 5 Summary
1D Optimization
I can …
- Find the local maxima, local minima, global maximum and global minimum for the graph of a function.
- Find the critical points of \(f(x)\) by solving \(f'(x)=0\).
- Use the first derivative test to determine whether a critical point is a local minimum, a local maximum or a point of inflection.
- Use the second derivative test to determine whether a critical point is a local minimum or a local maximum.
2D Optimization
I can …
- Find the 2D critical points of \(f(x,y)\) by (simultaneously) solving \(f_x(x,y)=0\) and \(f_y(x,y)=0\).
- Characterize a 2D critical point \((a,b)\) as a local minimimum, local maximum or saddle point by:
- Looking at the values of \(f(a,b)-f(x,y)\) on a small circle around \((a,b)\).
- Making a contour plot near \((a,b)\)
- Using the 2D second derivative test at \((a,b)\)
Gradient Search
I can …
- Find a local maximum of a 2D function \(f(x,y)\) using gradient search in RStudio.
- Find a local minimum of a 2D function \(f(x,y)\) using gradient search in RStudio.
- Simulate gradient search using the contour plot of a 2D function to find the trajectory that gradient search follows to reach an extreme point.
- Explain why the extreme point found by gradient search depends upon the starting point that we use.
Constrained Optimization
I can …
- Identify the objective function and the constaint function for a constrained optimization problem.
- Explain why the extreme point for a constrained optimization problem occurs where the constraint contour is tangent to a contour of the objective function.
- Use RStudio to create a contour plot to solve a constrained optimization problem.
- Estimate the value of the Lagrange multiplier \(\lambda\) using a contour plot.
- Interpret the Lagrange multiplier \(\lambda\) as a rate of change.