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A critical point (or stationary point) of f(x) is a point (a;f(a)) such that f0(a) = 0. Recall that, geometrically, these are points on the graph of f(x) who have a \ at" tangent line, i.e. a constant tangent line. Critical Points f(x) Example 1: Find all critical points of f(x) = x3 3x2 9x+ 5. We see that the derivative is f0(x) = 3x2 6x 9.
Lecture 10. Optimization problems for multivariable functions. Local maxima and minima - Critical points. (Relevant section from the textbook by Stewart: 14.7) Our goal is to now find maximum and/or minimum values of functions of several variables, e.g., f(x, y) over prescribed domains.
To check if a critical point is maximum, a minimum, or a saddle point, using only the first derivative, the best method is to look at a graph to determine the kind of critical point. For some applications we want to categorize the critical points symbolically.
This time we’ll introduce another new concept to open up a whole set of applications: using derivatives to find maxima and minima of functions. On the worksheet, we saw how to solve an example of this type of problem: we found the highest point on the graph of y = f(x) = x2 − x4. y. (− 1√ , 1 4) 2. −1.
Describe how to use critical points to locate absolute extrema over a closed interval. Given a particular function, we are often interested in determining the largest and smallest values of the function. This information is important in creating accurate graphs.
Example: Find the critical points of z = 1 − x2 − y 2 . ∂z ∂z. Answer: = −2x and = −2y. Clearly the only point where both derivatives are ∂x ∂y. 0 is (0, 0). Thus, there is a single critical point at (0, 0). The figure shows it is clearly the point where z reaches a maximum value.
Problem 11.1: Find all critical points for the following functions. If there are in nitely many, indicate their structure. For f(x) = cos(x) for example, the critical points can be written as ˇ=2 + kˇ, where kis an integer. a) f(x) = x6 3x2. b) f(x) = 4sin(ˇx) + 3 c) f(x) = exp( x2)x2. d) f(x) = sin(cos(ˇx))
