Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
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### Problem 5: Cost Function Analysis
The total cost function for a product is given by:
\[ C(x) = 800 \ln(x + 10) + 1900 \]
where \( x \) is the number of units produced.
**Tasks:**
(a) Calculate the total cost of producing 200 units. Round your answer to the nearest cent.
- **Answer Field:** \$ ______
(b) Determine how many units must be produced for the total costs to equal $7500. Round your answer to the nearest whole number.
- **Answer Field:** ______ units
**Guidance Available**
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### Problem 6: Continuous Compound Interest
For an investment of $9500 at an interest rate of 13.3% compounded continuously, the future value \( S \) at any time \( t \) in years is determined by the formula:
\[ S = 9500 e^{0.133t} \]
**Tasks:**
(a) Find the amount after 18 months. Round your answer to two decimal places.
- **Answer Field:** \$ ______
(b) Calculate how long it will take for the investment to double. Round your answer to two decimal places.
- **Answer Field:** ______ years
**Guidance Available**
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