Let $A = [{a_{ij}}]$ be a 3 $\times$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {if\,\left| {i - j} \right| = 1} \cr {2x + 1} & , & {otherwise.} \cr } } \right.$Let a function f : R $\to$$ R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to:

Question

Let $A = [{a_{ij}}]$ be a 3 $\times$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {if\,\left| {i - j} \right| = 1} \cr {2x + 1} & , & {otherwise.} \cr } } \right.$

Let a function f : R $\to$$ R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to:

$ - {{20} \over {27}}$

${{88} \over {27}}$

${{20} \over {27}}$

$ - {{88} \over {27}}$

Solution

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