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:
Solution
$$A = \left[ {\matrix{
1 & { - x} & {2x + 1} \cr
{ - x} & 1 & { - x} \cr
{2x + 1} & { - x} & 1 \cr
} } \right]$$<br><br>$\left| A \right| = 4{x^3} - 4{x^2} - 4x = f(x)$<br><br>$f'(x) = 4(3{x^2} - 2x - 1) = 0$<br><br>$\Rightarrow x = 1;x = {{ - 1} \over 3}$<br><br>$\therefore$ $f(1) = - 4;f\left( { - {1 \over 3}} \right) = {{20} \over {27}}$<br><br>Sum $= - 4 + {{20} \over 7} = - {{88} \over {27}}$
About this question
Subject: Mathematics · Chapter: Application of Derivatives · Topic: Tangents and Normals
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