For `f:R rarr R, f(x)=x^(4)-8x^(3)+22x^(2)-24x`, the sum of all local extreme value of f(x) is equal to

Options

A. `-9`
B. `-8`
C. `-17`
D. `6`

Answer

To find the sum of all local extreme values of the function \( f(x) = x^4 - 8x^3 + 22x^2 - 24x \), we will follow these steps:

### Step 1: Find the derivative of \( f(x) \)
We start by differentiating \( f(x) \) with respect to \( x \):

\[ f'(x) = \frac{d}{dx}(x^4 - 8x^3 + 22x^2 - 24x) \]

Using the power rule for differentiation:

\[ f'(x) = 4x^3 - 24x^2 + 44x - 24 \]

### Step 2: Set the derivative equal to zero
To find the critical points, we set the derivative equal to zero:

\[ 4x^3 - 24x^2 + 44x - 24 = 0 \]

### Step 3: Simplify the equation
We can simplify the equation by dividing all terms by 4:

\[ x^3 - 6x^2 + 11x - 6 = 0 \]

### Step 4: Factor the polynomial
Now we will try to factor the cubic polynomial. We can use the Rational Root Theorem to test possible rational roots. Testing \( x = 1 \):

\[ 1^3 - 6(1^2) + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 \]

Since \( x = 1 \) is a root, we can factor \( (x - 1) \) out of the polynomial. We can perform synthetic division or polynomial long division to factor it:

\[ x^3 - 6x^2 + 11x - 6 = (x - 1)(x^2 - 5x + 6) \]

### Step 5: Factor the quadratic
Next, we factor the quadratic \( x^2 - 5x + 6 \):

\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 6: Write the complete factorization
Thus, we can write the complete factorization of \( f'(x) \):

\[ f'(x) = (x - 1)(x - 2)(x - 3) = 0 \]

### Step 7: Find the critical points
Setting each factor equal to zero gives us the critical points:

\[ x = 1, \quad x = 2, \quad x = 3 \]

### Step 8: Evaluate \( f(x) \) at the critical points
Now we will evaluate \( f(x) \) at these critical points to find the local extreme values:

1. For \( x = 1 \):
\[ f(1) = 1^4 - 8(1^3) + 22(1^2) - 24(1) = 1 - 8 + 22 - 24 = -9 \]

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

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

### Step 9: Sum the local extreme values
Now we sum the local extreme values:

\[ f(1) + f(2) + f(3) = -9 + (-8) + (-9) = -26 \]

### Final Answer
The sum of all local extreme values of \( f(x) \) is \( -26 \).

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