First slide

Introduction to Determinants

Question

Suppose $a, b, c > 1$ and $f (x) = |\begin{array}{ccc} a^{- x} & a^{x} & x \\ b^{- 3 x} & b^{3 x} & 3 x^{3} \\ c^{- 5 x} & c^{5 x} & 5 x^{5} \end{array}|, x \in R$ then $f$ is

Moderate

A

a constant function
B

a polynomial of degree 5
C

an odd function
D

an even function

Solution

$f (- x) = |\begin{array}{ccc} a^{x} & a^{- x} & - x \\ b^{3 x} & b^{- 3 x} & - 3 x^{3} \\ c^{5 x} & c^{- 5 x} & - 5 x^{5} \end{array}|$
$= (- 1) (- 1) |\begin{array}{ccc} a^{- x} & a^{x} & x \\ b^{- 3 x} & b^{3 x} & 3 x^{3} \\ c^{- 5 x} & c^{5 x} & 5 x^{5} \end{array}|$
$= f (x)$
Thus, $f$ is an even function.

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