Find the maximum or minimum value of the function. \[ f(x)=-\frac{x^{2}}{3}+2 x+6 \] Is this a maximum or minimum value? \( \bigcirc \) maximum value minimum value

To find the maximum or minimum value of the function \( f(x)=-\frac{x^{2}}{3}+2x+6 \), we first note that the coefficient of \( x^2 \) is negative, which indicates that the parabola opens downward. Hence, there will be a maximum point. To find the \( x \)-coordinate of the vertex (where the maximum occurs), we use the formula \( x = -\frac{b}{2a} \), where \( a = -\frac{1}{3} \) and \( b = 2 \). Calculating this gives us: \[ x = -\frac{2}{2 \cdot -\frac{1}{3}} = 3. \] Substituting \( x = 3 \) back into the function: \[ f(3) = -\frac{(3)^2}{3} + 2 \cdot 3 + 6 = -3 + 6 + 6 = 9. \] Thus, the maximum value of the function is \( 9 \). To ensure your solution is error-free, always double-check your work and make sure to substitute correctly. Keep in mind that operations in algebra can get tricky, especially with signs and fractions. Write everything out step-by-step to avoid missing any details!