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Let f: Rvec satisfying |f(x)|lt=x^2AAx i...

Let `f: Rvec` satisfying `|f(x)|lt=x^2AAx in R` be differentiable at `x=0.` The find `f^(prime)(0)dot`

Text Solution

Verified by Experts

Since, `|f(x)|lex^(2),AA x in R" …(1)"`
`At" "x=0, |f(0)|le0`
`rArr" "f(0)=0" ...(2)"`
`"Now "f(0)=underset(hrarr0)lim(f(h)-f(0))/(h)=(underset(hrarr0)lim(f(h))/(h)" ...(3)"`
From (1), `|f(h)|le|h|^(2)`
`rArr" "(|f(h)|)/(|h|)le|h|`
`rArr" "underset(hrarr0)lim|(f(h))/(h)|leunderset(hrarr0)lim|h|`
`rArr" "|underset(hrarr0)lim(f(h)-0)/(h)|le0`
`rArr" " |f'(0)|le0`
`rArr" "f'(0)=0`
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