Question
If $$f\left( x \right) = 4x - {x^2},\,x\, \in \,R,$$ then $$f\left( {a + 1} \right) - f\left( {a - 1} \right)$$ is equal to :
A.
$$2\left( {4 - a} \right)$$
B.
$$4\left( {2 - a} \right)$$
C.
$$4\left( {2 + a} \right)$$
D.
$$2\left( {4 + a} \right)$$
Answer :
$$4\left( {2 - a} \right)$$
Solution :
$$\eqalign{
& {\text{Given,}}\,f\left( x \right) = 4x - {x^2} \cr
& \Rightarrow f\left( {a + 1} \right) - f\left( {a - 1} \right) \cr
& = \left\{ {4\left( {a + 1} \right) - {{\left( {a + 1} \right)}^2}} \right\} - \left\{ {4\left( {a - 1} \right) - {{\left( {a - 1} \right)}^2}} \right\} \cr
& = 8 - \left\{ {{{\left( {a + 1} \right)}^2} - {{\left( {a - 1} \right)}^2}} \right\} \cr
& = 8 - 4a \cr
& = 4\left( {2 - a} \right) \cr} $$