Find-min-value-f-x-x-4-x-5-x-6-x-7-

Question Number 170566 by cortano1 last updated on 27/May/22

Find \min value

f (x) = (x + 4) (x + 5) (x + 6) (x + 7)

Answered by som(math1967) last updated on 27/May/22

(x + 4) (x + 7) (x + 5) (x + 6)

(x^{2} + 11 x + 28) (x^{2} + 11 x + 30)

(a + 28) (a + 30) [l e t a = x^{2} + 11 x]

a^{2} + 58 a + 840

{(a + 29)}^{2} - 29^{2} + 840

{(a + 29)}^{2} - 1

{(a + 29)}^{2} ⩾ 0

{(a + 29)}^{2} - 1 ⩾ - 1

∴ (x + 4) (x + 5) (x + 6) (x + 7) ⩾ - 1

∴ m i n v a l u e o f f (x) i s - 1

Answered by ajfour last updated on 27/May/22

l e t x + \frac{11}{2} = t

f (t) = (t - \frac{3}{2}) (t - \frac{1}{2}) (t + \frac{1}{2}) (t + \frac{3}{2})

= (t^{2} - \frac{9}{4}) (t^{2} - \frac{1}{4})

l e t t^{2} - \frac{5}{4} = z

f (z) = z^{2} - 1

f (z) ∣_{m i n} = - 1

w h e n t = x + \frac{11}{2} = \pm \frac{\sqrt{5}}{2}

x = \frac{- 11 \pm \sqrt{5}}{2}

Commented by Tawa11 last updated on 08/Oct/22

Great sir

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