What is optimization?
In calculus, an optimization problem is one that is a word problem where one must find the highest or lowest possible quantity. Typically, it involves maximizing or minimizing functions or variables.
Steps to solving Optimization problems:
1. Identify all unknown variables and such
2. If not already found in step 1, identify the quantity in need of max/minimizing
3. If need be, draw a diagram
4. Explicitly define all unknowns using any different desired variables.
5. Create the algebraic equation(s) that involve your variable(s). You must isolate the variable that represents your quantity that needs to be max.minimized, defined by only one other variable. (i.e., Q = f(x), Q is optimized, x is the related variable.)
6. Identify the domain of function f(x).
7. Under consideration of defined domain, determine the absolutely/global maximum or maximum for f(x).
[source: BL]
2. If not already found in step 1, identify the quantity in need of max/minimizing
3. If need be, draw a diagram
4. Explicitly define all unknowns using any different desired variables.
5. Create the algebraic equation(s) that involve your variable(s). You must isolate the variable that represents your quantity that needs to be max.minimized, defined by only one other variable. (i.e., Q = f(x), Q is optimized, x is the related variable.)
6. Identify the domain of function f(x).
7. Under consideration of defined domain, determine the absolutely/global maximum or maximum for f(x).
[source: BL]
Key Terms:
Critical Point: For any function f, a point p in the domain of f where f'(p) = 0 or f'(p) is undefined is called a critical point of the function f.
Critical Value: A critical value of f is the value, f(p), or the output, at a critical point, p. Inflection Point: A point at which the graph of a function changes concavity. At the inflection point f"(p) is either zero or undefined, and the function's second derivative changes signs. Local Maximum: A function f has a local maximum at p if f(p) is greater than or equal to the values of f for points near p. Local Minimum: A function f has a local maximum at p if f(p) is less than or equal to the values of f for points near p. Global Maximum: The single greatest value of a function f over a specified domain. A function f has a global maximum at p if f(p) is greater than r equal to all values of f. Global Minimum: The single greatest value of a function f over a specified domain. A function f has a global minimum at p if f(p) is less than r equal to all values of f. Local Extrema and Critical Points: If a function has a local max or min that is not an endpoint on an interval, then the derivative at that point equals 0, and that point is a critical point. WARNING: Each local max or min is a critical point, but not every critical point is a local max or min. Inflection Points and Local Maxima and Minima of Derivatives: A function f with a continuous derivative has an inflection point p if either of the following conditions hold: • f' has a local max or local min at p • f" changes signs at p Global Maxima and Minima: • To find the global max and min of a continuous function on a closed interval, compare the values of the function at all critical points in the interval and at the endpoints. • To find the global max or min on an open interval, find the value of the function at all critical points and sketch or examine the graph of the function. Focus on the function values as x approaches + or - ∞. |
[source: BL]
First and Second Derivative Tests for Local Maxima and Minima
First Derivative
• If f'(x) changes from positive to negative at a C.P. then f(x) has a local maxima.
• If f'(x) changes from - to +, then f(x) has a local minimum
• If f(x) has a C.P. but derivative doesn't change, it has neither
Second Derivative
• If f"(x) is + at a C.P., then f(x) has a local min at a C.P.
• If f"(x) is - at a C.P., then f(x) has a local max at that C.P.
• If f'(x) changes from positive to negative at a C.P. then f(x) has a local maxima.
• If f'(x) changes from - to +, then f(x) has a local minimum
• If f(x) has a C.P. but derivative doesn't change, it has neither
Second Derivative
• If f"(x) is + at a C.P., then f(x) has a local min at a C.P.
• If f"(x) is - at a C.P., then f(x) has a local max at that C.P.
[source: BL]
practice problem
A new deal between apple and Blackmagic Design has opened where Blackmagic just released The Da Vinci Resolve 15, dramatically skipping of 13 and 14 for some unknown reason and has agreed to sell it with the brand new MacBooklet SuperPro Oxygen already installed. The revenue from selling "d" MacBooklets is given by equation (a) below, and the total cost of the MacBooklets is provided by equation (b) below. What is the quantity that will maximize the new partners' profits?
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