Answer :
The function g(x) = -2[tex]x^{2}[/tex] + 10x + 2 has a maximum value of 14.5. It does not have a minimum value of -35.5 or -14.5.
To determine whether the function g(x) = -2[tex]x^{2}[/tex] + 10x + 2 has a minimum or a maximum, we need to analyze its quadratic form. The general form of a quadratic function is f(x) = a[tex]x^{2}[/tex] + bx + c, where a, b, and c are constants. In this case, a = -2, b = 10, and c = 2.
Since the coefficient of the [tex]x^{2}[/tex] term (a) is negative (-2), we know that the parabola opens downwards, indicating a maximum point. To find the x-coordinate of the maximum point, we can use the formula x = -b / (2a). Plugging in the values, we have x = -10 / (2 * -2) = -10 / -4 = 5/2 = 2.5.
To find the corresponding y-coordinate, we substitute this value of x into the function: g(2.5) = -2[tex](2.5)^2[/tex] + 10(2.5) + 2 = -2(6.25) + 25 + 2 = -12.5 + 25 + 2 = 14.5.
Therefore, the function g(x) = -2[tex]x^{2}[/tex] + 10x + 2 has a maximum value of 14.5 at x = 2.5.
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