A) b = 0, c = 0, a is any real
B) a = 0, b = 0, c is any real
C) c = 0, a = 0, b is any real
D) a + b =0
Correct Answer: D
Solution :
[d]: \[f'({{0}^{-}})=\underset{h\to 0}{\mathop{\lim }}\,\frac{a\sinh +b{{e}^{h}}+c{{h}^{3}}-b}{-h}\] \[=-(a+b)\] \[f'({{0}^{+}})=\underset{h\to 0}{\mathop{\lim }}\,\frac{a\sinh +b{{e}^{h}}+c{{h}^{3}}-b}{h}=a+b\] \[\because \] f(x) is differentiable at x = 0 \[\therefore \] \[a+b=-(a+b)\Rightarrow (a+b)=0.\]
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