+86 Find the minimum value of \(x(x + 1)(x + 2)(x + 3),\) over all real numbers \(x.\)
+180 We can rewrite the expression as (x2+3x)(x+2)(x+3). This can be further factored as (x2+3x)(x2+5x+6).
Now we can consider two cases:
Case 1: x^2 + 3x >= 0
In this case, both x2 and 3x are non-negative. The product (x2+3x)(x2+5x+6) is minimized when x2+5x+6 is minimized. We can complete the square on x2+5x+6 to get (x+25)2+47. Since (x+25)2 is always non-negative, the minimum value in this case is 7/4.
Case 2: x^2 + 3x < 0
In this case, both x2 and 3x are non-positive. The product (x2+3x)(x2+5x+6) is minimized when x2+5x+6 is maximized. We can again complete the square on x2+5x+6 to get (x+25)2+47. Since (x+25)2 is always non-negative, the maximum value in this case is also 7/4.
We see that the minimum value is achieved regardless of whether x2+3x is positive or negative. Therefore, the minimum value of x(x+1)(x+2)(x+3) is 7/4.
