Let `f(x)` be a nonzero function whose all successive derivative exist and are nonzero. If `f(x), f' (x) and f''(x)` are in G.P. and `f (0) = 1, f '(0) = 1`, then -

Options

A. `f(x) gt 0 forall X in R`
B. `f'(x) gt 0 forall X in R`
C. `f'''(0) = 1`
D. `f(x) le 1 forall X in R`

Answer

To solve the problem step by step, we start with the given conditions and use the properties of derivatives and geometric progression (G.P.).

### Step 1: Understand the condition of G.P.
Given that \( f(x), f'(x), f''(x) \) are in G.P., we can use the property of G.P. which states that if \( a, b, c \) are in G.P., then \( b^2 = ac \). Therefore, we can write:
\[ (f'(x))^2 = f(x) \cdot f''(x) \]

### Step 2: Rewrite the equation
Let’s denote \( f(x) \) as \( y \). Then, we have:
\[ (f'(x))^2 = y \cdot f''(x) \]

### Step 3: Change of variables
We can express \( f'(x) \) as \( \frac{dy}{dx} \) and \( f''(x) \) as \( \frac{d^2y}{dx^2} \). Thus, we can rewrite the equation as:
\[ \left( \frac{dy}{dx} \right)^2 = y \cdot \frac{d^2y}{dx^2} \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ \frac{d^2y}{dx^2} = \frac{\left( \frac{dy}{dx} \right)^2}{y} \]

### Step 5: Separation of variables
Now we can separate variables:
\[ \frac{d^2y}{dx^2} = \frac{(dy)^2}{y} \]
This can be rewritten as:
\[ \frac{dy}{dx} \cdot \frac{dy}{y} = d\left( \frac{dy}{dx} \right) \]

### Step 6: Integrate both sides
Integrating both sides gives:
\[ \int \frac{dy}{y} = \int \frac{d^2y}{dx^2} \, dx \]
This leads to:
\[ \log(y) = \frac{dy}{dx} + C_1 \]

### Step 7: Substitute initial conditions
We know \( f(0) = 1 \) and \( f'(0) = 1 \). Substituting these values:
- When \( x = 0 \), \( y = 1 \) implies \( \log(1) = 0 \).
- Thus, \( C_1 = 0 \).

### Step 8: Solve for \( y \)
This implies:
\[ \log(y) = \frac{dy}{dx} \]
Exponentiating both sides gives:
\[ y = e^{\frac{dy}{dx}} \]

### Step 9: Differentiate again
Differentiating both sides:
\[ \frac{dy}{dx} = e^{\frac{dy}{dx}} \cdot \frac{d^2y}{dx^2} \]

### Step 10: Solve the differential equation
This leads us to the conclusion that:
\[ f(x) = e^x \]

### Step 11: Verify the conditions
We verify:
- \( f(0) = e^0 = 1 \)
- \( f'(0) = e^0 = 1 \)
- \( f''(0) = e^0 = 1 \)

Thus, all conditions are satisfied.

### Conclusion
The function \( f(x) = e^x \) satisfies all given conditions.