If `y = f(x)` and `y^(4) - 4y + x = 0`. If `f (-8) = 2`, then the value of `|28 f' (- 8)|` is

Answer

To solve the problem step-by-step, we start with the given equation and the information provided.

### Step 1: Write down the given equation
We have the equation:
\[ y^4 - 4y + x = 0 \]
where \(y = f(x)\).

### Step 2: Substitute \(y = f(x)\)
Substituting \(y\) with \(f(x)\), we rewrite the equation as:
\[ f(x)^4 - 4f(x) + x = 0 \]

### Step 3: Differentiate the equation with respect to \(x\)
Differentiating both sides with respect to \(x\) using implicit differentiation:
\[ \frac{d}{dx}(f(x)^4) - \frac{d}{dx}(4f(x)) + \frac{d}{dx}(x) = 0 \]
Using the chain rule, we get:
\[ 4f(x)^3 f'(x) - 4f'(x) + 1 = 0 \]

### Step 4: Factor out \(f'(x)\)
Rearranging the equation, we can factor out \(f'(x)\):
\[ f'(x)(4f(x)^3 - 4) + 1 = 0 \]
This implies:
\[ f'(x)(4f(x)^3 - 4) = -1 \]

### Step 5: Solve for \(f'(x)\)
We can express \(f'(x)\) as:
\[ f'(x) = \frac{-1}{4f(x)^3 - 4} \]

### Step 6: Substitute \(x = -8\) and \(f(-8) = 2\)
We know that \(f(-8) = 2\). Substituting \(x = -8\) into the equation for \(f'(x)\):
\[ f'(-8) = \frac{-1}{4(2)^3 - 4} \]

### Step 7: Calculate \(f'(-8)\)
Calculating the denominator:
\[ 4(2)^3 = 4 \times 8 = 32 \]
So,
\[ 4(2)^3 - 4 = 32 - 4 = 28 \]
Thus,
\[ f'(-8) = \frac{-1}{28} \]

### Step 8: Calculate \(|28 f'(-8)|\)
Now we need to find:
\[ |28 f'(-8)| = |28 \times \frac{-1}{28}| \]
This simplifies to:
\[ | -1 | = 1 \]

### Final Answer
Thus, the value of \(|28 f'(-8)|\) is:
\[ \boxed{1} \]