यदि `f(x) = (x-1) (x-2) (x-3)` यदि "a = 0, b = 4 , तो माध्यमान प्रमेय से ' c ' का मान ज्ञात कीजिए ।

Explanation

` :' f(x) = ( x-1) ( x-2) (x-3) `
` = x^(3) - 6x^(2) + 11x - 6 `
` :. F(0) = -6`
`f(4) = (4)^(3) - 6 (4)^(2) + 11 (4) -6`
=6
` f'(x) = 3x^(2) - 12x + 11 `
` :. F'(c ) = 3c^(2) - 12c + 11`
अब माध्यमान प्रमेय से ,
` f' (c ) = ( f(b) - f(a) )/(b-a) = ( 6-(-6))/(4-0) `
` 3c^(2) - 12c + 11 = 3 `
` 3c^(2) - 12c + 11 - 3 = 0 `
` 3c^(2) - 12c + 8 = 0 `
` c = ( 12 pm sqrt( 144- 4 xx 3 xx 8 ))/(2 xx3) `
` = ( 12 pm sqrt( 48) )/6 = ( 12 pm 4sqrt(3))/6 = 2 pm (2sqrt(3))/3`