Question

Y=x^2-10x+64/x^2+10x+64 find the minimum value of y

Answer 1

we have to find the minimum value of y = (x² - 10x + 64)/(x² + 10x + 64)

⇒y = (x² - 10x + 64)/(x² + 10x + 64)

⇒yx² + 10yx + 64y = x² - 10x + 64

⇒(y - 1)x² + 10x(y + 1) + 64(y - 1) = 0

now discriminant = b² - 4ac ≥ 0

⇒{10(y + 1)² - 4 × 64(y - 1) × (y - 1) ≥ 0

⇒100(y + 1)² - 256(y - 1)² ≥ 0

⇒{10(y + 1)}² - {16(y - 1)}² ≥ 0

⇒(10y + 10 - 16y + 16)(10y + 10 + 16y - 16) ≥ 0

⇒(-6y + 26)(26y - 6) ≥ 0

⇒3/13 ≤ y ≤ 13/3

therefore, minimum value of y is 3/13

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Answer 2

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