a) Find the Cost Function (C(x))? b) When the function (C(1) the fixed constant (K $22 ), at the initial condition (x ?), and rewrite the (C(x)) equation, with the 1); Find fixed constant known. Hint: Logarithmic Rule: In(1) = 0 C(x) = [| 48x³ 7 dx + 40
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.
A company’s Marginal Cost function is ()x(MC) is given below; where (x) is the number of units produced:
![### Calculus Problem: Finding the Cost Function
#### Problem Statement:
**a)** Find the Cost Function \( C(x) \)?
**b)** When the function \( C(1) = \$22 \), at the initial condition \( x = 1 \); Find the fixed constant \( K = ? \), and rewrite the \( C(x) \) equation, with the fixed constant known.
**Hint:** Logarithmic Rule: \(\ln(1) = 0\)
#### Given:
\[
C(x) = \int \left(48x^3 - 40 + \frac{7}{x} \right) dx
\]
#### Solution:
**Step 1: Integrate the Given Function**
First, perform the integration of the given function:
\[
C(x) = \int \left(48x^3 - 40 + \frac{7}{x}\right) dx
\]
**Step 2: Applying Integration Rules**
1. For the integral of \(48x^3\):
\[
\int 48x^3 dx = 48 \cdot \frac{x^4}{4} = 12x^4
\]
2. For the integral of \(-40\):
\[
\int -40 dx = -40x
\]
3. For the integral of \(\frac{7}{x}\):
\[
\int \frac{7}{x} dx = 7 \ln|x|
\]
**Step 3: Combining Results**
After integration, combine these results:
\[
C(x) = 12x^4 - 40x + 7 \ln|x| + K
\]
Here, \( K \) is the constant of integration.
**Step 4: Find the Fixed Constant \( K \)**
Use the initial condition \( C(1) = 22 \):
\[
22 = 12(1)^4 - 40(1) + 7 \ln(1) + K
\]
Knowing that \( \ln(1) = 0 \):
\[
22 = 12 - 40 + 0 + K
\]
\[
22 = -28 + K
\]
\[
K = 50
\]
**Step 5: Write the Final Cost Function**
Substitute \( K = 50 \) back into the cost function:
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc8d17bc-eeef-4f5c-bae1-870d59d58eb9%2Fcd6adbce-8716-4ea2-90ed-3fe4821bc53e%2Fc4lomlp.png&w=3840&q=75)
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