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# 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
6 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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