First slide

Monotonicity

Question

If a, b, c are real numbers, then the intervals in which $f (x) = |\begin{array}{l} x + a^{2} a b a c \\ a b x + b^{2} b c \\ a c b c x + c^{2} \end{array}| is strictly decreasing$

Moderate

A

$x < 0$
B

$x > 0$
C

$0 < x < 1$
D

$[- \frac{2}{3}, 0]$

Solution

$\begin{aligned} f (x) = \frac{1}{a b c} |\begin{array}{ccc} a (x + a^{2}) & a b^{2} & a c^{2} \\ a^{2} b & b (x + b^{2}) & b c^{2} \\ a^{2} c & b^{2} c & c (x + c^{2}) \end{array}| \\ = \frac{a b c}{a b c} |\begin{array}{ccc} x + a^{2} & b^{2} & c^{2} \\ a^{2} & x + b^{2} & c^{2} \\ a^{2} & b^{2} & x + c^{2} \end{array}| \\ = |\begin{array}{ccc} x + a^{2} + b^{2} + c^{2} & b^{2} & c^{2} \\ x + a^{2} + b^{2} + c^{2} & x + b^{2} & c^{2} \\ x + a^{2} + b^{2} + c^{2} & b^{2} & x + c^{2} \end{array}| \\ = (x + a^{2} + b^{2} + c^{2}) |\begin{array}{ccc} 1 & b^{2} & c^{2} \\ 1 & x + b^{2} & c^{2} \\ 1 & b^{2} & x + c^{2} \end{array}| \end{aligned}$
$\begin{aligned} = x^{2} (x + a^{2} + b^{2} + c^{2}) \\ f^{'} (x) = 3 x^{2} + 2 x (a^{2} + b^{2} + c^{2}) \\ f^{'} (x) < 0 \\ x (3 x + 2) (a^{2} + b^{2} + c^{2}) \leq 0 \\ - \frac{2}{3} (a^{2} + b^{2} + c^{2}) \leq x \leq 0 \end{aligned}$
$\begin{aligned} x (3 x + 2) \leq 0 \\ - \frac{2}{3} \leq x \leq 0 \end{aligned}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App