Question

The minimum value of the polynomial p(x) = 3x^2 - 5x + 2 is ?in kannada

Answer 1

$Given \: \: Question \: \: Is \: \\ find \: minimum \: \: of \: p(x) \: if \\ \\ p(x) = 3x {}^{2} - 5x + 2 \\ \\ Answer \: \\ \\ \: \: 3x {}^{2} - 5x + 2 = p(x) \\ \\ Multiply \: \: both \: sides \: by \: \: \frac{1}{3} \\ \\ x {}^{2} - \frac{5x}{3} + \frac{2}{3} = \frac{p(x)}{3} \\ \\ x {}^{2} - 2(1)( \frac{5x}{6} ) + \frac{5 {}^{2} }{6 {}^{2} } - \frac{5 {}^{2} }{6 {}^{2} } + \frac{2}{3} = \frac{p(x)}{3} \\ \\ (x - \frac{5}{6} ) {}^{2} + \frac{2}{3} - \frac{5 {}^{2} }{6 {}^{2} } = \frac{ p(x)}{3} \\ \\ (x - \frac{5}{6} ) {}^{2} + \frac{72 - 75}{3 \times 6 {}^{2} } = \frac{p(x)}{3} \\ \\ (x - \frac{5}{6} ) {}^{2} - \frac{1}{6 {}^{2} } = \frac{p(x)}{3} \\ \\Logically \: \: \: \\ \\ p(x) \: \: will \: \: be \: \: minimum \: \: when \: \: (x - \frac{5}{6} ) {}^{2} \\ is \: \: minimum \: \\ And \: \\ the \: minimum \: \: value \: \: of \: \: (x - \frac{5}{6} ) {}^{2} \: \: is \: \: zero \\ \\ 0 - \frac{1}{6 {}^{2} } = \frac{p(x)}{3} \\ \\ p(x) = - \frac{3}{6 {}^{2} } \\ \\ p(x) = - \frac{1}{12} \\ \\ therefore \: \: the \: minimum \: value \: of \:\\ p(x) = 3x {}^{2} - 5x + 2 \: \: is \\ \\ - \frac{1}{12}$