A function 
𝑓
(
𝑥
)
f(x) which is the best fit for the data given in the Table-1 recorded by a student, is 
𝑓
(
𝑥
)
=
−
(
𝑥
−
1
)
2
(
𝑥
−
3
)
(
𝑥
−
5
)
(
𝑥
−
7
)
+
𝑐
f(x)=−(x−1) 
2
 (x−3)(x−5)(x−7)+c What will be the value of 
𝑐
c, so that SSE (Sum Squared Error) will be minimum?

To find the value of \( c \) that minimizes the Sum of Squared Errors (SSE) for the given function \( f(x) = - (x-1)^2 (x-3)(x-5)(x-7) + c \), we'll go through the steps to compute the SSE and then find the optimal \( c \).

### Given Data
\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
y & 4 & 18 & 4 & -24 & 3 \\
\hline
\end{array}
\]

### Step-by-Step Solution:

1. **Define the function \( f(x) \):**
   \[
   f(x) = - (x-1)^2 (x-3)(x-5)(x-7) + c
   \]

2. **Compute \( f(x) \) at the given \( x \) points:**
   \[
   \begin{aligned}
   f(1) &= - (1-1)^2 (1-3)(1-5)(1-7) + c = c \\
   f(2) &= - (2-1)^2 (2-3)(2-5)(2-7) + c = -1 \cdot (-1) \cdot (-3) + c = 3 + c \\
   f(3) &= - (3-1)^2 (3-3)(3-5)(3-7) + c = 0 + c = c \\
   f(4) &= - (4-1)^2 (4-3)(4-5)(4-7) + c = - 3^2 (1)(-1)(-3) + c = - 9 \cdot (-3) + c = 27 + c \\
   f(5) &= - (5-1)^2 (5-3)(5-5)(5-7) + c = - 4^2 (2)(0) + c = 0 + c = c
   \end{aligned}
   \]

3. **Compute the errors \( e_i = f(x_i) - y_i \) and SSE:**
   \[
   \begin{aligned}
   e_1 &= c - 4 \\
   e_2 &= 3 + c - 18 = c - 15 \\
   e_3 &= c - 4 \\
   e_4 &= 27 + c + 24 = c + 51 \\
   e_5 &= c - 3 \\
   \end{aligned}
   \]

   The SSE is given by:
   \[
   \text{SSE} = (c - 4)^2 + (c - 15)^2 + (c - 4)^2 + (c + 51)^2 + (c - 3)^2
   \]

4. **Simplify SSE and minimize it:**
   \[
   \begin{aligned}
   \text{SSE} &= (c - 4)^2 + (c - 15)^2 + (c - 4)^2 + (c + 51)^2 + (c - 3)^2 \\
   &= 3(c - 4)^2 + (c - 15)^2 + (c + 51)^2 + (c - 3)^2 \\
   &= 3(c^2 - 8c + 16) + (c^2 - 30c + 225) + (c^2 + 102c + 2601) + (c^2 - 6c + 9) \\
   &= 3c^2 - 24c + 48 + c^2 - 30c + 225 + c^2 + 102c + 2601 + c^2 - 6c + 9 \\
   &= 6c^2 + c^2 + c^2 + c^2 + (-24c - 30c + 102c - 6c) + (48 + 225 + 2601 + 9) \\
   &= 6c^2 + c^2 + c^2 + c^2 + (-24c - 30c + 102c - 6c) + (48 + 225 + 2601 + 9) \\
   &= 9c^2 + 42c + 2883
   \end{aligned}
   \]

   To minimize SSE, we set the derivative with respect to \( c \) to zero:

   \[
   \frac{d}{dc}(9c^2 + 42c + 2883) = 18c + 42 = 0
   \]

   Solving for \( c \):

   \[
   18c + 42 = 0 \implies 18c = -42 \implies c = -\frac{42}{18} = -\frac{7}{3}
   \]

Therefore, the value of \( c \) that minimizes the SSE is \( c = -\frac{7}{3} \).

Would you like further details on any specific part of the solution or have any questions?

Here are some related questions you might find interesting:

1. How do we derive the function for SSE in terms of \( c \)?
2. What is the role of the sum of squared errors (SSE) in regression analysis?
3. Can you show the plot of the given function and the data points?
4. How does changing \( c \) affect the fit of the function to the data points?
5. What other methods can be used to find the best fit for a given set of data?

**Tip:** When working with regression models, always verify the fit by plotting the function alongside your data points to visually inspect the accuracy of the fit.

Learn how to minimize the Sum of Squared Errors (SSE) for a polynomial function given specific data points. This problem involves finding the optimal constant 'c' in a polynomial function to minimize errors, using derivatives and SSE formula. Suitable for advanced high school and college students.

Math Problem Statement

A function 𝑓 ( 𝑥 ) f(x) which is the best fit for the data given in the Table-1 recorded by a student, is 𝑓 ( 𝑥 )

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