Let `f (x)=x+ (x ^(2))/(2 )+ (x ^(3))/(3 )+ (x ^(4))/(4 ) +(x ^(5))/(5)` and let `g (x) = f ^(-1) (x).` Find `g''(o).`

Answer

To find \( g''(0) \) where \( g(x) = f^{-1}(x) \) and \( f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} \), we can follow these steps:

### Step 1: Understanding the relationship between \( f \) and \( g \)
Since \( g(x) = f^{-1}(x) \), we have:
\[ f(g(x)) = x \]
Differentiating both sides with respect to \( x \):
\[ f'(g(x)) \cdot g'(x) = 1 \]
Thus, we can express \( g'(x) \) as:
\[ g'(x) = \frac{1}{f'(g(x))} \]

### Step 2: Finding \( g(0) \)
To find \( g(0) \), we need to determine \( f^{-1}(0) \). We need to find \( x \) such that \( f(x) = 0 \):
\[ f(0) = 0 + 0 + 0 + 0 + 0 = 0 \]
Thus, \( g(0) = f^{-1}(0) = 0 \).

### Step 3: Finding \( g'(0) \)
Now substituting \( x = 0 \) into the expression for \( g'(x) \):
\[ g'(0) = \frac{1}{f'(g(0))} = \frac{1}{f'(0)} \]

### Step 4: Finding \( f'(x) \)
Now we need to compute \( f'(x) \):
\[ f'(x) = 1 + x + x^2 + x^3 + x^4 \]
Evaluating at \( x = 0 \):
\[ f'(0) = 1 + 0 + 0 + 0 + 0 = 1 \]
Thus:
\[ g'(0) = \frac{1}{1} = 1 \]

### Step 5: Finding \( g''(x) \)
To find \( g''(x) \), we differentiate \( g'(x) \):
\[ g''(x) = -\frac{f''(g(x)) \cdot (g'(x))^2}{(f'(g(x)))^2} \]

### Step 6: Finding \( g''(0) \)
Now substituting \( x = 0 \):
\[ g''(0) = -\frac{f''(g(0)) \cdot (g'(0))^2}{(f'(g(0)))^2} \]
Since \( g(0) = 0 \):
\[ g''(0) = -\frac{f''(0) \cdot (g'(0))^2}{(f'(0))^2} \]

### Step 7: Finding \( f''(x) \)
Now we compute \( f''(x) \):
\[ f''(x) = 1 + 2x + 3x^2 + 4x^3 \]
Evaluating at \( x = 0 \):
\[ f''(0) = 1 + 0 + 0 + 0 = 1 \]

### Step 8: Putting it all together
Now substituting the values we found:
\[ g''(0) = -\frac{1 \cdot (1)^2}{(1)^2} = -1 \]

Thus, the final answer is:
\[ \boxed{-1} \]