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Let f be an even function and let f ’ (0) exist . then find f ‘ (0)

Let f be an even function and let f ’ (0) exist . 
then find f ‘ (0) 

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 years ago
Ans:
f(x) = f(-x)
f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}
f’(0) exist. So
LHD (x=0) = RHD (x=0)
\lim_{h\rightarrow 0}\frac{f(-h)-f(0)}{-h} = \lim_{h\rightarrow 0}\frac{f(h)-f(0)}{h}
f(-h) = f(h)
\lim_{h\rightarrow 0}\frac{f(h)-f(0)}{-h} = \lim_{h\rightarrow 0}\frac{f(h)-f(0)}{h}
Both are zero by zero form. So apply L’Hospital rule,
\lim_{h\rightarrow 0}-f'(h) = \lim_{h\rightarrow 0}f'(h)
-f'(0) = f'(0)
f'(0) = 0
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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