Let f be a real valued, twice differentiable function on R such that \underset{t \textrm{ } \rightarrow \textrm{ } x}{Lim} \textrm{ } \frac{f^{3} \left(\right. t \left.\right) \textrm{ } - \textrm{ } f^{3} \left(\right. x \left.\right)}{t^{3} - x^{3}} =

Engineering

Mathematics

L Hospital Rule

Question

Let f be a real valued, twice differentiable function on R such that $\underset{t \to x}{Lim} \frac{f^{3} (t) - f^{3} (x)}{t^{3} - x^{3}}$ = – 1, f(0) = 1. Then

f(f(100)) = 100

$\int_{- 1}^{1} f^{3} (x) dx = 2$

f(x) = f ^–1(x)

f(x) is strictly increasing function

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$\underset{t \to x}{Lim} \frac{f^{3} (t) - f^{3} (x)}{t^{3} - x^{3}} = - 1 \Rightarrow \frac{3 f^{2} (x) f' (x)}{3 x^{2}} - 1 \Rightarrow$ f ²(x) f '(x) = – x²
$\Rightarrow f' (x) \leq 0 \forall x \in R$
Now integrating $\frac{f^{3} (x)}{3} = \frac{- x^{3}}{3} + C$
f(x) = (k – x³)^1/3
f(0) = 1
$∴$ k = 1
f(x) = (1 – x³)^1/3 ⇒ f(f(x)) = x.