what-is-range-function-y-x-1-5-x-

Question Number 85176 by jagoll last updated on 19/Mar/20

what is range

function y = \sqrt{x - 1} + \sqrt{5 - x}

Answered by john santu last updated on 19/Mar/20

Domain : x ⩾ 1 \land x ⩽ 5 \Rightarrow 1 ⩽ x ⩽ 5

Range : y^{'} = \frac{1}{2 \sqrt{x - 1}} - \frac{1}{2 \sqrt{5 - x}} = 0

\sqrt{x - 1} = \sqrt{5 - x} \Rightarrow x - 1 = 5 - x

2 x = 3 \Rightarrow x = 3 .

y_{\max} = \sqrt{2} + \sqrt{2} = 2 \sqrt{2}

for x = 1 \Rightarrow y = \sqrt{5 - 1} = 2

for x = 5 \Rightarrow y = \sqrt{5 - 1} = 2

range : 2 ⩽ y ⩽ 2 \sqrt{2}

Commented by jagoll last updated on 19/Mar/20

thanks you

Answered by mr W last updated on 19/Mar/20

y = \sqrt{x - 1} + \sqrt{5 - x} ⩾ 0

y^{2} = 4 + 2 \sqrt{(x - 1) (5 - x}) ⩾ 4

\Rightarrow y ⩾ 2 \dots (i)

\frac{y^{2}}{2} - 2 = \sqrt{(x - 1) (5 - x)}

{(\frac{y^{2}}{2} - 2)}^{2} = (x - 1) (5 - x)

x^{2} + 6 x + 5 + {(\frac{y^{2}}{2} - 2)}^{2} = 0

D = 6^{2} - 4 (5 + {(\frac{y^{2}}{2} - 2)}^{2}) ⩾ 0

4 - {(\frac{y^{2}}{2} - 2)}^{2} ⩾ 0

- 2 ⩽ \frac{y^{2}}{2} - 2 ⩽ 2

0 ⩽ \frac{y^{2}}{2} ⩽ 4

\Rightarrow y ⩽ 2 \sqrt{2} \dots (i i)

f r o m (i) a n d (i i) w e g e t r a n g e :

2 ⩽ y ⩽ 2 \sqrt{2}

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