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if f(x) and g(x) are continuous function...

if `f(x)` and `g(x)` are continuous functions, `fog` is identity function, `g'(b) = 5` and `g(b) = a` then `f'(a)` is

A

(a)`2/5`

B

(b)`1/5`

C

(c)`3/5`

D

(d)5

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the given information step by step. ### Step 1: Understand the given information We know that \( f(g(x)) = x \) is the identity function. This means that applying \( g \) followed by \( f \) returns the original input \( x \). ### Step 2: Differentiate both sides Differentiating both sides with respect to \( x \), we apply the chain rule: \[ \frac{d}{dx}[f(g(x))] = \frac{d}{dx}[x] \] This gives us: \[ f'(g(x)) \cdot g'(x) = 1 \] ### Step 3: Substitute the known values We are given that \( g(b) = a \) and \( g'(b) = 5 \). We can substitute \( x = b \) into our differentiated equation: \[ f'(g(b)) \cdot g'(b) = 1 \] Substituting \( g(b) = a \) and \( g'(b) = 5 \): \[ f'(a) \cdot 5 = 1 \] ### Step 4: Solve for \( f'(a) \) Now, we can isolate \( f'(a) \): \[ f'(a) = \frac{1}{5} \] ### Conclusion Thus, the value of \( f'(a) \) is \( \frac{1}{5} \). ### Final Answer The answer is \( \frac{1}{5} \). ---
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