If a2-b2-c2-1 and x2-y2-z2-1- where a- b- c- x- y- z - 0- then the maximum value of a x-b y-c z is

The correct option is B 1As we know, A.M. ≥ G.M., a^2+x^22≥ ax......(1) b^2+y^22≥ by......(2) c^2+z^22≥ cz......(3) Adding (1),(2) and (3) ax+by+cz ≤a^2+x^22+ ...

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Byju's Answer

Standard XII

Mathematics

Derivative from First Principle

If a2+b2+c2=1...

Question

If $a^{2} + b^{2} + c^{2} = 1$ and $x^{2} + y^{2} + z^{2} = 1,$ where $a, b, c, x, y, z \geq 0,$ then the maximum value of $a x + b y + c z$ is

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Solution

The correct option is B $1$
As we know, A.M. $\geq$ G.M.,
$\frac{a^{2} + x^{2}}{2} \geq a x . . . . . . (1)$
$\frac{b^{2} + y^{2}}{2} \geq b y . . . . . . (2)$
$\frac{c^{2} + z^{2}}{2} \geq c z . . . . . . (3)$

Adding $(1), (2)$ and $(3)$
$a x + b y + c z \leq \frac{a^{2} + x^{2}}{2} + \frac{b^{2} + y^{2}}{2} + \frac{c^{2} + z^{2}}{2}$
$\Rightarrow a x + b y + c z \leq 1$

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If a2+b2+c2=1 and x2+y2+z2=1, where a,b,c,x,y,z≥0, then the maximum value of ax+by+cz is

The correct option is B 1As we know, A.M. ≥ G.M., a2+x22≥ax......(1) b2+y22≥by......(2) c2+z22≥cz......(3) Adding (1),(2) and (3) ax+by+cz≤a2+x22+b2+y22+c2+z22 ⇒ax+by+cz≤1

If $a^{2} + b^{2} + c^{2} = 1$ and $x^{2} + y^{2} + z^{2} = 1,$ where $a, b, c, x, y, z \geq 0,$ then the maximum value of $a x + b y + c z$ is