If the function `f (x)=-4e^((1-x)/(2))+1 +x+(x ^(2))/(2)+ (x ^(3))/(3) and g (x) =f ^(-1)(x),` then the value of `g'((-7)/(6))` equals :

Options

A. `1/5`
B. `-1/5`
C. `6/7`
D. `-6/7`

Answer

To find the value of \( g'(-\frac{7}{6}) \) where \( g(x) = f^{-1}(x) \) and \( f(x) = -4e^{\frac{1-x}{2}} + 1 + x + \frac{x^2}{2} + \frac{x^3}{3} \), we can follow these steps:

### Step 1: Understand the relationship between \( f \) and \( g \)
Since \( g(x) = f^{-1}(x) \), we know that:
\[ f(g(x)) = x \]
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: Find \( g(-\frac{7}{6}) \)
To find \( g'(-\frac{7}{6}) \), we first need to find \( g(-\frac{7}{6}) \), which is equivalent to finding \( f^{-1}(-\frac{7}{6}) \). This means we need to solve the equation:
\[ f(x) = -\frac{7}{6} \]

### Step 3: Set up the equation
We need to solve:
\[ -4e^{\frac{1-x}{2}} + 1 + x + \frac{x^2}{2} + \frac{x^3}{3} = -\frac{7}{6} \]
Rearranging gives:
\[ -4e^{\frac{1-x}{2}} + 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{7}{6} = 0 \]
This simplifies to:
\[ -4e^{\frac{1-x}{2}} + x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{13}{6} = 0 \]

### Step 4: Guess and check for a solution
We can try \( x = 1 \):
\[ f(1) = -4e^{\frac{1-1}{2}} + 1 + 1 + \frac{1^2}{2} + \frac{1^3}{3} \]
Calculating this gives:
\[ f(1) = -4 \cdot 1 + 1 + 1 + \frac{1}{2} + \frac{1}{3} = -4 + 1 + 1 + 0.5 + 0.333 = -4 + 2.833 = -1.167 \]
This does not equal \(-\frac{7}{6}\).

Now, let's try \( x = 0 \):
\[ f(0) = -4e^{\frac{1-0}{2}} + 1 + 0 + 0 + 0 = -4e^{0.5} + 1 \]
Calculating \( e^{0.5} \approx 1.6487 \):
\[ f(0) \approx -4 \cdot 1.6487 + 1 \approx -6.5948 + 1 = -5.5948 \]
This also does not equal \(-\frac{7}{6}\).

### Step 5: Finding the correct \( x \)
After testing values, we find that \( x = 1 \) satisfies the equation \( f(1) = -\frac{7}{6} \). Thus:
\[ g(-\frac{7}{6}) = 1 \]

### Step 6: Find \( f'(x) \)
Now we need to compute \( f'(x) \):
\[ f'(x) = \frac{d}{dx} \left(-4e^{\frac{1-x}{2}} + 1 + x + \frac{x^2}{2} + \frac{x^3}{3}\right) \]
Calculating the derivative:
\[ f'(x) = -4 \cdot \left(-\frac{1}{2} e^{\frac{1-x}{2}}\right) + 1 + x + x^2 \]
Simplifying:
\[ f'(x) = 2e^{\frac{1-x}{2}} + 1 + x + x^2 \]

### Step 7: Evaluate \( f'(1) \)
Now we need to evaluate \( f'(1) \):
\[ f'(1) = 2e^{0} + 1 + 1 + 1 = 2 + 1 + 1 + 1 = 5 \]

### Step 8: Find \( g'(-\frac{7}{6}) \)
Now substituting back to find \( g'(-\frac{7}{6}) \):
\[ g'(-\frac{7}{6}) = \frac{1}{f'(g(-\frac{7}{6}))} = \frac{1}{f'(1)} = \frac{1}{5} \]

### Final Answer
Thus, the value of \( g'(-\frac{7}{6}) \) is:
\[ \boxed{\frac{1}{5}} \]