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Find the maximum and minimum values, if any, of the following functions given by(i) f (x) = |x + 2| – 1(ii) g(x) = – | x + 1| + 3(iii) h(x) = sin (2x) + 5(iv) f (x) = |sin 4x + 3|(v) h(x) = x + 1, x ∈ (- 1, 1) |
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Answer» (i) f(x) = |x + 2| – 1 f(x) = |x + 2| – 1 ≥ -1 minimum value is – 1 when x + 2 = 0, x = -2 however it has no maximum value. (ii) g(x) = – | x + 1| + 3 = 3,-1 x + 1| g(x) < 3 v x + 1 = 0 ∴ max. value is 3 when x = – 1 how ever no minimum value. (iii) h(x) = sin (2x) + 5 maximum value of sin 2x = 1 and minimum value is -1 f(x) = 1 + 5 is 1 + 5 ∴ max. h (x) = 6 and min. h (x) = 4. (iv) f(x) = |sin 4x + 3| -1 < sin 4x < 1 ⇒ 3 -1 < sin 4x + 3 < + 1 + 3 + 2′< sin 4 x + 3 < 4 2 < | sin 4x + 3 | < 4 f(x) > 2 and f (x) < 4 min. f(x) = 2 when sin 4x + 3 = 0 max f (x) = 4 when sin 4x + 3 = 0 ∴ minimum value is 2 at sin 4x = -1 maximum value is 4 at sin 4x = 1 (v) h(x) = x + 1, x ∈ (-1,1) given that x ∈ (-1, 1) i.e. -1 < x < 1 -1 + 1< x + 1 < 1 + 10 < x + 1< 2 ∴ x + 1 > 0 or x + 1 < 2 x + 1 > 0 so no minimum value x + 1 < 2, so no maximum value. |
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