To solve the given problem, let's analyze the function step-by-step.
### Part (a)
Consider the function [tex]\( f(x) = -3x^2 + 6x - 4 \)[/tex]. This is a quadratic function in the form of [tex]\( f(x) = ax^2 + bx + c \)[/tex] where:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- [tex]\( c = -4 \)[/tex]
For a quadratic function, if the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a \)[/tex]) is positive ([tex]\( a > 0 \)[/tex]), the parabola opens upwards and the function has a minimum value. If [tex]\( a \)[/tex] is negative ([tex]\( a < 0 \)[/tex]), the parabola opens downwards and the function has a maximum value.
Since [tex]\( a = -3 \)[/tex] is less than 0, the function has a maximum value.
- a. The function has a maximum value.
### Part (b)
To find the maximum value of the function and where it occurs, we locate the vertex of the parabola. The vertex of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
For the given function:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = 6 \)[/tex]
The [tex]\( x \)[/tex]-coordinate of the vertex is:
[tex]\[ x = -\frac{b}{2a} = -\frac{6}{2(-3)} = 1 \][/tex]
To find the corresponding [tex]\( f(x) \)[/tex]-value (the maximum value), we substitute [tex]\( x = 1 \)[/tex] back into the function:
[tex]\[ f(1) = -3(1)^2 + 6(1) - 4 \][/tex]
[tex]\[ f(1) = -3 + 6 - 4 \][/tex]
[tex]\[ f(1) = -1 \][/tex]
Therefore, the maximum value of the function is [tex]\( -1 \)[/tex] and it occurs at [tex]\( x = 1 \)[/tex].
### Part (c)
Domain:
The domain of any quadratic function is all real numbers because you can substitute any real number for [tex]\( x \)[/tex] and get a real number output. So, the domain is:
[tex]\[ \text{all real numbers} \][/tex]
Range:
Since the function opens downwards (as [tex]\( a < 0 \)[/tex]), the range is all real numbers less than or equal to the maximum value. The highest point (maximum value) of the function is [tex]\( -1 \)[/tex].
Thus, the range can be written as:
[tex]\[ \text{Range: } (-\infty, -1] \][/tex]
### Summary:
a. The function has a maximum value.
b. The maximum value is [tex]\( -1 \)[/tex] and it occurs at [tex]\( x = 1 \)[/tex].
c. The domain of the function is all real numbers, and the range is [tex]\( (-\infty, -1] \)[/tex].
