Thank you for registering.

One of our academic counsellors will contact you within 1 working day.

Please check your email for login details.
MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping

Show that if f is in?nitely di?erentiable on (-1, 1) and f(1/n)= 0 for all n > 1 then f^(k)(0) = 0 for all k > 0. [f^(k)(x) is differentiating f(x) k times]

Show that if f is in?nitely di?erentiable on (-1, 1) and f(1/n)= 0 for all n > 1 then f^(k)(0) = 0 for all k > 0. [f^(k)(x) is differentiating f(x) k times]

Grade:

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
6 years ago
Ans:
f is infinitely differentiable on (-1, 1). So it would be continuous in (-1, 1).
f(\frac{1}{n}) = 0, n> 1
\lim_{n\rightarrow \infty }f(\frac{1}{n}) = f(0) = 0
f'(x) = \lim_{h\rightarrow 0}\frac{f(h) - f(0)}{h}
f'(0) = \lim_{h\rightarrow 0}\frac{f(h) }{h}
h is the delta neibhbourhood of ‘0’. So its value is zero.
f'(0) = \lim_{h\rightarrow 0}\frac{0 }{h} = 0
f''(0) = \lim_{h\rightarrow 0}\frac{f'(h) - f'(0)}{h}
f''(0) = \lim_{h\rightarrow 0}\frac{f'(h)}{h}
f''(0) = \lim_{h\rightarrow 0}\frac{0}{h} = 0
Similarly for the kthderivative,
f^{k}(0) = \lim_{h\rightarrow 0}\frac{f^{k-1}(h) - f^{k-1}(0)}{h}
f^{k}(0) = \lim_{h\rightarrow 0}\frac{f^{k-1}(h)}{h}
f^{k}(0) = \lim_{h\rightarrow 0}\frac{0}{h} = 0
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

Think You Can Provide A Better Answer ?

Provide a better Answer & Earn Cool Goodies See our forum point policy

ASK QUESTION

Get your questions answered by the expert for free