Question:
If \(f(x)\) is a real valued bijective function and twice differentiable function. If \(g(x)\) is inverse of \(f(x)\) and \(f(0)=a\), then \(g''(a)=\)
If \(f(x)\) is a real valued bijective function and twice differentiable function. If \(g(x)\) is inverse of \(f(x)\) and \(f(0)=a\), then \(g''(a)=\)
Show Hint
For an inverse function,
\[
\boxed{
(f^{-1})''(y)
=
-\frac{f''(x)}{[f'(x)]^3},
\qquad y=f(x).
}
\]
Always substitute the corresponding value of \(x\) after differentiation.
Updated On: Jul 18, 2026
- \[ -\frac{f''(0)}{[f'(0)]^3} \]
- \[ -\frac{f''(a)}{[f'(a)]^3} \]
- \[ \frac{f''(0)}{[f'(a)]^2} \]
- \[ -\frac{f''(a)}{[f'(0)]^2} \]
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Use the derivative of the inverse function.
If
\[
g=f^{-1},
\]
then
\[
g'(y)=\frac{1}{f'(x)},
\]
where
\[
y=f(x).
\]
Differentiating once again,
\[
g''(y)
=
-\frac{f''(x)}{[f'(x)]^3}.
\]
Step 2: Use the given condition. Since \[ f(0)=a, \] we have \[ g(a)=0. \] Hence, in the above formula, substitute \[ x=0. \] Therefore, \[ g''(a) = -\frac{f''(0)}{[f'(0)]^3}. \]
Step 3: Write the final answer. Hence, \[ \boxed{ g''(a) = -\frac{f''(0)}{[f'(0)]^3} }. \] Thus, \[ \boxed{(A)} \] is the correct answer.
Step 2: Use the given condition. Since \[ f(0)=a, \] we have \[ g(a)=0. \] Hence, in the above formula, substitute \[ x=0. \] Therefore, \[ g''(a) = -\frac{f''(0)}{[f'(0)]^3}. \]
Step 3: Write the final answer. Hence, \[ \boxed{ g''(a) = -\frac{f''(0)}{[f'(0)]^3} }. \] Thus, \[ \boxed{(A)} \] is the correct answer.
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