find-range-function-f-x-x-7x-x-2-1-without-calculus-

Question Number 83977 by john santu last updated on 08/Mar/20

find range function

f (x) = x \sqrt{7 x - x^{2} - 1} without

calculus

Answered by john santu last updated on 08/Mar/20

Answered by TANMAY PANACEA last updated on 08/Mar/20

x \times \sqrt{- x^{2} + 7 x - 1}

x \times \sqrt{- 1 - (x^{2} - 2 \times x \times \frac{7}{2} + \frac{49}{4} - \frac{49}{4})}

x \times \sqrt{- 1 + \frac{49}{4} - {(x - \frac{7}{2})}^{2}}

x \times \sqrt{\frac{45}{4} - {(x - \frac{7}{2})}^{2}}

{(x - \frac{7}{2})}^{2} ≯ \frac{45}{4}

\frac{45}{4} - {(x - \frac{7}{2})}^{2} ⩾ 0

(\frac{3 \sqrt{5}}{2} + x - \frac{7}{2}) (\frac{3 \sqrt{5}}{2} - x + \frac{7}{2}) ⩾ 0

c r i t i c a l v a l u e o f x = \frac{7 - 3 \sqrt{5}}{2} a n d \frac{7 + 3 \sqrt{5}}{2}

(x - a) (b - x) ⩾ 0 [a = \frac{7 - 3 \sqrt{5}}{2} b = \frac{7 + 3 \sqrt{5}}{2} b > a]

g (x) = (x - a) (b - x)

w h e n x > b g (x) < 0

w h e n x < a g (x) < 0

w h e n b > x > a g (x) > 0

g (x) = 0 w h e n x = a a n d x = b

f (x) = x \times \sqrt{- x^{2} + 7 x - 1}

f (x) = x \times \sqrt{(x - a) (b - x)} [b > a]

f (a) = f (b) = 0 b u t x ≯ b, x ≮ a

w a i t \dots

Commented by john santu last updated on 08/Mar/20

domain \frac{45}{4} - {(x - \frac{7}{2})}^{2} ⩾ 0

{(x - \frac{7}{2})}^{2} - {(\frac{3 \sqrt{5}}{2})}^{2} ⩽ 0

(x - \frac{7}{2} + \frac{3 \sqrt{5}}{2}) (x - \frac{7}{2} - \frac{3 \sqrt{5}}{2}) ⩽ 0

\frac{7 - 3 \sqrt{5}}{2} ⩽ x ⩽ \frac{7 + 3 \sqrt{5}}{2}

Commented by john santu last updated on 08/Mar/20

range y = \sqrt{- x^{4} + 7 x^{3} - x^{2}}

Commented by john santu last updated on 08/Mar/20

y^{'} = \frac{- 4 x^{3} + 21 x^{2} - 2 x}{2 \sqrt{- x^{4} + 7 x^{3} - x^{2}}} = 0

4 x^{3} - 21 x^{2} + 2 x = 0

x (4 x^{2} - 21 x + 2) = 0

x^{2} - \frac{21}{4} x + \frac{1}{2} = 0

{(x - \frac{21}{8})}^{2} + \frac{1}{2} - \frac{441}{64} = 0