Question

If f(x) = x³ + x − 2, find (f⁻¹)'(0).

Answer 1

f(x) = x³ + x – 2                        ...(1)

Differentiating w.r.t. x, we get

f'(x) = `"d"/"dx"(x^3 + x  – 2)`

= 3x² + 1 - 0

= 3x² + 1

We know that

`(f^-1) ^' (y) = (1)/(f"'"(x)`                        ...(2)  

From (1), y = f(x) = 0, when x = 0

∴ from (2), (f^-1)'(0)

= `(1)/(f"'"(0)) = 1/(3x^2+1)`

`(1)/(f"'"(0)) = 1/(3(0)^2 + 1)`           ....[∵ x = 0]

`(1)/(f"'"(0)) = 1/(0 + 1)`

`(1)/(f"'"(0)) = 1/1`

`(1)/(f"'"(0)) = 1`

∴ f'(0) = 1