Which of the following correctly describes the minimum value of f(x)=2x^2+12x+1 A Since f(x)=2(x+3)^2-8 the minimum value of f(x) is -3 B Since f(x)=2(x+3)^2-8 the minimum value of f(x) is -8 C Since f(x)=2(x+1)(x+5) the minimum value of f(x) is -1 D Since f(x)=2(x+1)(x+5) the minimum value of f(x) is -5

To find the minimum value of the quadratic function $f(x)=2x^{2}+12x+1$, we can complete the square to rewrite the function in vertex form.<br /><br />Completing the square:<br />$f(x)=2x^{2}+12x+1$<br />$=2(x^{2}+6x)+1$<br />$=2(x^{2}+6x+9-9)+1$<br />$=2(x+3)^{2}-18+1$<br />$=2(x+3)^{2}-17$<br /><br />The vertex form of the quadratic function is $f(x)=2(x+3)^{2}-17$.<br /><br />The vertex of the parabola is at the point $(-3, -17)$, which means the minimum value of $f(x)$ is $-17$.<br /><br />Therefore, the correct answer is:<br />B Since $f(x)=2(x+3)^{2}-17$ the minimum value of $f(x)$ is $-17$.