Let `f:R rarrR, f(x)=x^(4)-8x^(3)+22x^(2)-24x+c`.
If sum of all extremum value of f(x) is 1, then c is equal to

Options

A. 8
B. 9
C. 10
D. 11

Answer

To solve the problem, we need to find the value of \( c \) such that the sum of all extremum values of the function \( f(x) = x^4 - 8x^3 + 22x^2 - 24x + c \) equals 1.

### Step 1: Find the first derivative of \( f(x) \)
The first step is to differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^4 - 8x^3 + 22x^2 - 24x + c) \]
Calculating the derivative:
\[ f'(x) = 4x^3 - 24x^2 + 44x - 24 \]

**Hint:** Remember that the derivative gives us the slope of the function, and we set it to zero to find critical points.

### Step 2: Set the first derivative to zero
To find the extremum points, we set \( f'(x) = 0 \):
\[ 4x^3 - 24x^2 + 44x - 24 = 0 \]
Dividing the entire equation by 4 simplifies it:
\[ x^3 - 6x^2 + 11x - 6 = 0 \]

**Hint:** Look for rational roots using the Rational Root Theorem or synthetic division.

### Step 3: Factor the cubic equation
We can factor the cubic polynomial:
\[ x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3) \]
Thus, the critical points are:
\[ x = 1, \quad x = 2, \quad x = 3 \]

**Hint:** Verify the roots by substituting them back into the polynomial.

### Step 4: Calculate the extremum values
Next, we need to find the values of \( f(1) \), \( f(2) \), and \( f(3) \):
1. **Calculate \( f(1) \)**:
\[ f(1) = 1^4 - 8(1^3) + 22(1^2) - 24(1) + c = 1 - 8 + 22 - 24 + c = -9 + c \]

2. **Calculate \( f(2) \)**:
\[ f(2) = 2^4 - 8(2^3) + 22(2^2) - 24(2) + c = 16 - 64 + 88 - 48 + c = -8 + c \]

3. **Calculate \( f(3) \)**:
\[ f(3) = 3^4 - 8(3^3) + 22(3^2) - 24(3) + c = 81 - 216 + 198 - 72 + c = -9 + c \]

**Hint:** Substitute the values carefully and simplify to find the extremum values.

### Step 5: Sum the extremum values
Now, we sum the extremum values:
\[ f(1) + f(2) + f(3) = (-9 + c) + (-8 + c) + (-9 + c) = 3c - 26 \]

**Hint:** Ensure that you combine like terms correctly.

### Step 6: Set the sum equal to 1
According to the problem, the sum of all extremum values equals 1:
\[ 3c - 26 = 1 \]

**Hint:** Isolate \( c \) by adding 26 to both sides.

### Step 7: Solve for \( c \)
Now, solve for \( c \):
\[ 3c = 27 \implies c = 9 \]

Thus, the value of \( c \) is:
\[ \boxed{9} \]