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If a + b + c = 12 (where a, b, c are real numbers), then the minimum value of a2 + b2 + c2 is:
If a + b + c = 12 (where a, b, c are real numbers), then the minimum value of a2 + b2 + c2 is:
1). 96
2). 100
3). 98
4). 48
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a + b + c = 12
Squaring both sides we get,
(a + b + c)2 = 122
⇒ a2 + b2 + c2 + 2(ab + bc + ca) = 144 ….(i)
⇒ a2 + b2 + c2 = 144 – 2(ab + bc + ca)
Here a2 + b2 + c2 will be minimum only when (ab + bc + ca) will be maximum.
And for ab + bc + ca to be maximum, a,b,c must be equal.
⇒ a = b = c = 4
⇒ ab + bc + ca = 4×4 + 4×4 + 4×4 = 48
Putting this value in eqn (i) we get,
a2 + b2 + c2 + 2×48 = 144
⇒ a2 + b2 + c2 = 144 – 96 = 48
∴ a2 + b2 + c2 = 48