let `f (x) = int _(0) ^(x) e ^(x-y) f'(y) dy - (x ^(2) -x+1)e ^(x)`
Find the number of roots of the equation `f (x) =0.`

Answer

To solve the problem step by step, we start with the function defined by the integral:

Given:
\[ f(x) = \int_0^x e^{x-y} f'(y) \, dy - (x^2 - x + 1)e^x \]

We need to find the number of roots of the equation \( f(x) = 0 \).

### Step 1: Analyze the Integral
We can rewrite the integral part:
\[ \int_0^x e^{x-y} f'(y) \, dy \]

Using the substitution \( u = x - y \), we can express \( dy = -du \) and change the limits accordingly. However, we will directly use integration by parts for simplicity.

### Step 2: Integration by Parts
Let:
- \( u = e^{x-y} \) (first part)
- \( dv = f'(y) \, dy \) (second part)

Then:
- \( du = -e^{x-y} \, dy \)
- \( v = f(y) \)

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
We get:
\[ \int_0^x e^{x-y} f'(y) \, dy = \left[ e^{x-y} f(y) \right]_0^x - \int_0^x f(y)(-e^{x-y}) \, dy \]

### Step 3: Evaluate the Boundary Terms
Evaluating the boundary terms:
- At \( y = x \): \( e^{x-x} f(x) = f(x) \)
- At \( y = 0 \): \( e^{x-0} f(0) = e^x f(0) \)

Thus:
\[ \int_0^x e^{x-y} f'(y) \, dy = f(x) - e^x f(0) + \int_0^x f(y)e^{x-y} \, dy \]

### Step 4: Substitute Back into \( f(x) \)
Substituting back into \( f(x) \):
\[ f(x) = f(x) - e^x f(0) + \int_0^x f(y)e^{x-y} \, dy - (x^2 - x + 1)e^x \]

### Step 5: Simplifying the Equation
Rearranging gives:
\[ e^x f(0) + (x^2 - x + 1)e^x = \int_0^x f(y)e^{x-y} \, dy \]

### Step 6: Setting Up the Equation
Now we need to find the roots of:
\[ f(x) = 0 \]
This leads to:
\[ e^x (f(0) + (x^2 - x + 1)) = \int_0^x f(y)e^{x-y} \, dy \]

### Step 7: Analyze the Roots
We know that \( e^x \) is never zero. Therefore, we need to analyze:
\[ f(0) + (x^2 - x + 1) = 0 \]

### Step 8: Finding \( f(0) \)
From the original equation, we can find \( f(0) \):
When \( x = 0 \):
\[ f(0) = \int_0^0 e^{0-y} f'(y) \, dy - (0^2 - 0 + 1)e^0 = 0 - 1 = -1 \]

Thus:
\[ -1 + (x^2 - x + 1) = 0 \]
This simplifies to:
\[ x^2 - x + 0 = 0 \]

### Step 9: Solving the Quadratic
The quadratic equation is:
\[ x^2 - x = 0 \]
Factoring gives:
\[ x(x - 1) = 0 \]

### Step 10: Finding the Roots
The roots are:
\[ x = 0 \quad \text{and} \quad x = 1 \]

### Conclusion
Thus, the number of roots of the equation \( f(x) = 0 \) is:
**2 roots: \( x = 0 \) and \( x = 1 \)**.