Answer:
It has a minimum in the point (3, 5)
Step-by-step explanation:
A quadratic function always has either a minimum or a maximum - never both. To find out which one, just look at the coefficient at the x^2. If it's positive, it has a minimum (think of its value for very low x - the value gets higher and higher, because x^2 is positive and rising the lower x is - similar for very high xs when x increases - so it must have a smallest point in the middle).
Analogously, if the coefficient at the x^2 is negative, the reasoning above is reversed - so it has to have a highest point.
Because out function has a positive coefficient at x^2 (the coefficient is 1), we know the function has a minimum.
BTW, let's find the minimum. We can rewrite the function as
[tex]f(x) = x^2 - 6x + 14 = x^2 - 6x + 9 + 5 = (x - 3)^2 + 5[/tex]
Notice that you can do that to any quadratic function by analyzing the ratio between coefficient at x^2 and at x.
Now the function is an x^2 function that was moved 5 upwards and 3 to the right. The minimum has to be in x - 3 = 0, so x = 3, and the value is 5!
The minimum is at (3, 5).
