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  4. The value of (1+Δ) (1- nabla) is

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Q. The value of $ \left(1+\Delta\right) \left(1- \nabla\right) $ is

MHT CETMHT CET 2011

Solution:

We have, $(1+\Delta)(1-\nabla) f(x) $
$=(1+\Delta)\{(1-\nabla) f(x)\} $
$= (1+\Delta)\{f(x)-\nabla f(x)\} $
$= (1+\Delta)[f(x)-\{f(x)-f(x-h)\}] $
$= (1+\Delta) f(x-h)=E f(x-h)$
$[\because(E=1+\Delta)]$
$=f(x)=1 \cdot f(x)$
Thus, $(1+\Delta)(1-\nabla) f(x)=1 \cdot f(x),$
for any function $f(x)$
$\therefore (1+\Delta)(1+\nabla) =1$