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question mark

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

Drake , 11 Years ago
Grade 12
anser 1 Answers
Jitender Singh
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
Last Activity: 11 Years ago
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