Statement I if `f(0)=a,f'(0)=b,g(0)=0,(fog)'(0)=c` then `g'(0)=(c)/(b).` Statement II `(f(g(x))'=f'(g(x)).g'(x),` for all `n`

Options

A. Both statement I and Statement II are correct and Statement II is the correct explanation of Statement I
B. Both Statement I and Statement II are correct but Statement II is not the correct explanation of Statement I
C. Statement I is correct but Statement II is incorrect
D. Statement II is correct but Statement I is incorrect.

Answer

To solve the question, we will analyze both statements step by step.

### Step 1: Understanding Statement II
**Statement II**: \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \) for all \( x \).

This statement is a representation of the chain rule in differentiation. The chain rule states that if you have a composition of functions, the derivative of that composition is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

**Hint for Step 1**: Recall the chain rule for differentiation, which is essential for understanding how derivatives of composed functions work.

### Step 2: Applying Statement II at \( x = 0 \)
Now, we will apply this rule at \( x = 0 \).

1. Substitute \( x = 0 \) into the chain rule:
\[ (f(g(0)))' = f'(g(0)) \cdot g'(0) \]

2. From the problem, we know:
- \( g(0) = 0 \)
- Therefore, \( f(g(0)) = f(0) = a \).

3. Thus, we can rewrite the equation:
\[ (f(0))' = f'(0) \cdot g'(0) \]
Since \( f(0) = a \), we have:
\[ 0 = f'(0) \cdot g'(0) \]

**Hint for Step 2**: Remember that the derivative of a constant (like \( f(0) \)) is zero.

### Step 3: Using Given Values
From the problem, we have:
- \( f'(0) = b \)
- Therefore, substituting this into the equation gives:
\[ 0 = b \cdot g'(0) \]

### Step 4: Solving for \( g'(0) \)
To isolate \( g'(0) \):
1. If \( b \neq 0 \), then the only solution to \( 0 = b \cdot g'(0) \) is:
\[ g'(0) = 0 \]

2. If \( b = 0 \), then \( g'(0) \) can be any value, but we need to consider the context of the problem.

### Step 5: Relating to Statement I
**Statement I**: Given \( (f \circ g)'(0) = c \).

From the chain rule, we have:
\[ (f \circ g)'(0) = f'(g(0)) \cdot g'(0) = f'(0) \cdot g'(0) = b \cdot g'(0) \]
Since we know \( (f \circ g)'(0) = c \), we can equate:
\[ b \cdot g'(0) = c \]

### Step 6: Solving for \( g'(0) \)
From the equation \( b \cdot g'(0) = c \), we can solve for \( g'(0) \):
\[ g'(0) = \frac{c}{b} \]

### Conclusion
Thus, we conclude that:
- Statement I is true: \( g'(0) = \frac{c}{b} \).
- Statement II is true as it represents the chain rule.

### Final Answer
Both statements are correct, and Statement II is the correct explanation of Statement I.