Question 203074
Functions of the form: g(x) = ax^2 + bx + c are (vertically oriented) parabolas. And I hope you know what parabolas look like. They are somewhat u-shaped (or upside-down u-shaped).<br>
Your function is of this form, with a = 2, b = 3 and c = 1. So your function is a parabola.<br>
The bottom of the "U" (or, for the upside-down ones, the top of the "U") is called the vertex. The vertex of a parabola with be the maximum or minimum value, depending on whether the "U" is right-side up or upside-down.<br>
So the problem you have is to:<ol><li>Find the vertex of the parabola.</li><li>Determine if the parabola is right-side up or upside down.</li><li>Use the answer of #2 to determine if the vertex is a maximum or minimum,</li></ol>
1. Find the vertex. The vertex of the parabola will be when the x-value is -b/(2a). Your b is 3 and your a is 2 so the x-value of the vertex will be {{{-b/(2a) = -(3)/(2*(2)) = -3/4}}}. Now we use this x-value to find the function value (the y-value) for the vertex: {{{f(-3/4) = 2(-3/4)^2 + 3(-3/4) + 1 = 2(9/16) + 3(-3/4) + 1 = 18/16 + (-9/4) + 1 = 18/16 + (-36/16) + 16/16 = -2/16 = -1/8}}}. So the vertex is (-3/4, -1/8).<br>
2. Determine if the parabola is right-side up or upside-down. If a > 0 then the parabola is right-side up. If a < 0 then the parabola is upside-down. Your a is 2 so your parabola is right-side up.<br>
3. Determine if the vertex is a maximum or a minimum value. Since the parabola is right-side up, the vertex is the bottom of the "U". So the vertex is a minimum value.<br>
The minimum value for f(x) is -1/8 (when x = -3/4).