First slide

Differentiability

Question

$If f : R \to R such that |f (x + y) - f (x - y) - 2 y^{3} - 6 x^{2} y| \leq y^{4} \forall x, y \in R then$

Difficult

A

Number of points of inflection of f(x) is 1
B

Number of critical points is 2
C

Number of local maxima are 2
D

Number of local minimum are zero

Solution

$Put x + y = u; x - y = v$
$\begin{array}{l} ∣ f (u) - f (v) - \{(x + y)^{3} - (x - y)^{3}\} ∣\leq {(\frac{u - v}{2})}^{4} \\ |f (u) - f (v) - \{u^{3} - v^{3}\}| \leq {(\frac{u - v}{2})}^{4} \\ | g (u) - g (v) | \leq {(\frac{u - v}{2})}^{4}, (g (x) = f (x) - x^{3}) \end{array}$
$\lim_{u \to v} | \frac{g (u) - g (v)}{u - v} | \leq \lim_{u \to v} {(\frac{u - v}{2})}^{3} = 0 |g^{'} (x)| \leq 0 \Rightarrow g^{'} (x) = 0 \Rightarrow g (x) i s a c o n s \tan t f u n c t i o n . l e t i t b e k . t h e r e f o r e f (x) = x^{3} + k (1) f^{'} (x) = 3 x^{2} = 0 \Rightarrow x = 0, w h i c h i s p o i n t o f i n f l e c t i o n$

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