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Question
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
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Solution
We have,
`f (x) = 9x^2 + 12x + 2 = 9 (x^2 + 4/3 x) + 2`
`= 9 {x^2 + 4/3x + 4/9} + 2 - 4 = 9 (x + 2/3)^2 - 2`
Since, `(x + 2/3)^2 >= 0`
= `9 (x + 2/3)^2 - 2 >= -2`
= f (x) ≥ -2 for all x ∈ R.
∴ Minimum f (x) = -2, which occurs when,
`x + 2/3 = 0, i.e, when (x + 2/3) = 0` when `x = -2/3`
f (x) has no maximum value, for f (x), f (x) → ∞ as |x| → ∞
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