Question

determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and find the value. f(x)= - 3x^2 + 6x - 1 does the quadratic function f have a minimum value or a maximum value? the function f has a minimum value. the function f has a maximum value. what is this minimum or maximum value? - 5 (simplify your answer.)

Explanation:

Step1: Identify coefficients

For the quadratic function $f(x)=-3x^{2}+6x - 1$, $a=-3$, $b = 6$, $c=-1$. Since $a=-3<0$, the parabola opens downwards and the function has a maximum value.

Step2: Find x - coordinate of vertex

The x - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. Substitute $a=-3$ and $b = 6$ into the formula: $x=-\frac{6}{2\times(-3)} = 1$.

Step3: Find maximum value

Substitute $x = 1$ into the function $f(x)=-3x^{2}+6x - 1$. Then $f(1)=-3\times1^{2}+6\times1 - 1=-3 + 6-1=2$.

Answer:

2