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## Differential Calculus

### Course: Differential Calculus>Unit 5

Lesson 11: Solving optimization problems

# Optimization: sum of squares

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.B (LO)
,
FUN‑4.B.1 (EK)
,
FUN‑4.C (LO)
,
FUN‑4.C.1 (EK)
What is the minimum possible value of x^2+y^2 given that their product has to be fixed at xy = -16. Created by Sal Khan.

## Want to join the conversation?

• Well, but what is optimization ? •   Optimization just means finding the value that maximizes or minimizes something. In this case, we optimized (minimized) the sum of two numbers squared.
• What does calculus represent? What does it do for us? •   Our world is in constant motion, or change, so if you want to describe/explain/model/predict this world, the best tool to use is one that was developed to describe processes that change over time, namely the calculus; poetic license can share the emotive/descriptive aspect of change, but not tell us anything about how or why. Almost everything you see around you in the world today, internet, computers, highways, bridges, buildings, successful business, planes, trains and automobiles, medicines and medical devices, TVs, cell phones and even computer games could not be at the level of complexity they have now with out calculus.
• When you solve x^2=16 you said it's 4. But this could be -4 too, and so Y would be 4 instead of -4. • At : Wherefore do you take the second derivative of the original equation? • When in real life would we use this?
and when would we use any other calculus stuff? • If the graph is always concave upwards (since Y''(x) is always positive) how can there be two minimums (at 4 and -4)? • Graphically, If x = +4 and -4 than function will have 2 points where derivative of f(x) = 0, means there will be two points where sope of f(x) = 0 . Where is the second point if Funcion is always concave upwards and doesn't have an inflection point ? • The function has zero slope at x=4 and x=-4. It doesn't have an inflection point, but it has an asymptote at x=0 because one of the terms has x in the denominator and tends to infinity. It may appear to be a curve that's always concave up, but actually it's two separate curves that are concave up, with a discontinuity at x=0.
• Couldn't we just input a value in the first derivative that is close to x=4 from both the positive and negative sides. If the first derivatives gives a negative value for the side of x that is more negative, and the first derivative gives a value that is positive for the side of x that is more positive, then don't we know that x=4 is a minimum value? b/c the original function's slope is negative approaching from negative infinity and positive approaching from positive infinity? • The IWDDSP method to solving optimization problems:

1. Identify the quantity to be maximized/minimized and all given values.
2. Write an equation in terms of a single variable for the quantity to be maximized/minimized.
3. Determine the domain of the function.
4. Differentiate both sides of the equation with respect to the variable in the equation.
5. Set the derivative equal to zero and solve for the values that could maximize/minimize the desired quantity.
6. Plug the resulting values, along with the endpoints of the domain, into the equation to find the maximum/minimum quantity. • Is the concept of maximisation and minimisation the same thing as convexity and concavity? • Not exactly, but they are related.

Concavity is a description of how the slope of a function changes, and finding concavity involves looking at the second derivative of the parent function for zeros, which correspond to possible inflection points.

Optimization problems are problems of identifying certain extrema, and tend to involve not just finding them (which would be just looking at the first derivative of the parent function for zeros, which correspond to possible critical points/extrema) but also describing the parent function in the first place, determining it from a worded description of the problem. This is shown in the video here, where the word problem "minimize the sum of the squares of two numbers whose product is -16" must be translated into "minimize S(x), the single-variable function which represents the sum of the squares of two numbers whose product is -16".

The two concepts are related, in that the extrema found in optimization problems are usually at points that look like the tops of crests (maxima) or the bottoms of troughs (minima), and as a result the concavity around those extrema is very visually distinguishable.

(Note that when I say "possible" points, I mean that for it to be "actually" a critical or inflection point, the first or second derivative (respectively) would have to have an actual sign change in the neighborhood of the zero.)