If a -2 b - c -4 and a - b - c - R- then the maximum value of ab - bc - ca is

Given nbsp;a+2b+c=4 ⇒ a=4 2b c Let nbsp;ab+bc+ca=y ⇒ a(b+c)+bc=y ⇒ (4 2b c)(b+c)+bc=y ⇒ 2b^2+2(c 2)b 4c+c^2+y=0 Since, b∈ℝ So, D≥0 ⇒ 4(c 2)^2 4×2( 4c+c^2+y ...

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Byju's Answer

Standard XIII

Mathematics

Nature of Roots

If a +2 b + c...

Question

If $a + 2 b + c = 4$ and $a, b, c \in R,$ then the maximum value of $(a b + b c + c a)$ is

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Solution

Given $a + 2 b + c = 4$
$\Rightarrow a = 4 - 2 b - c$
Let $a b + b c + c a = y$
$\Rightarrow a (b + c) + b c = y$
$\Rightarrow (4 - 2 b - c) (b + c) + b c = y$
$\Rightarrow 2 b^{2} + 2 (c - 2) b - 4 c + c^{2} + y = 0$

Since, $b \in R$
So, $D \geq 0$
$\Rightarrow 4 (c - 2)^{2} - 4 \times 2 (- 4 c + c^{2} + y) \geq 0$
$\Rightarrow c^{2} - 4 c + 2 y - 4 \leq 0$

Since, $c \in R$
So, $D \geq 0$
$\Rightarrow 16 - 4 (2 y - 4) \geq 0$
$\Rightarrow y \leq 4$

$∴ max (a b + b c + c a) = 4$

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If a+2b+c=4 and a,b,c∈R, then the maximum value of (ab+bc+ca) is

Given a+2b+c=4 ⇒a=4−2b−c Let ab+bc+ca=y ⇒a(b+c)+bc=y ⇒(4−2b−c)(b+c)+bc=y ⇒2b2+2(c−2)b−4c+c2+y=0 Since, b∈R So, D≥0 ⇒4(c−2)2−4×2(−4c+c2+y)≥0 ⇒c2−4c+2y−4≤0 Since, c∈R So, D≥0 ⇒16−4(2y−4)≥0 ⇒y≤4 ∴max(ab+bc+ca)=4

If $a + 2 b + c = 4$ and $a, b, c \in R,$ then the maximum value of $(a b + b c + c a)$ is