Question

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=3 x^{2}+24 x\)

Short answer

The function has a minimum value of -48 at x = -4.

Step-by-step solution

Identify the coefficient of the quadratic term

The function is given by \[f(x) = 3x^2 + 24x\]. Notice that the coefficient of the quadratic term (\[x^2\]) is 3.

Determine if the parabola opens upwards or downwards

Since the coefficient of \[x^2\] is positive (3 > 0), the parabola opens upwards. This means the quadratic function has a minimum value.

Find the vertex of the parabola

The vertex form of a quadratic function \[ax^2 + bx + c\] has its vertex at \[x = -\frac{b}{2a}\]. Here, \[a = 3\] and \[b = 24\]. Plugging in these values, we get \[x = -\frac{24}{2 \times 3} = -4\].

Calculate the minimum value

Substitute \[x = -4\] back into the function to find the minimum value: \[f(-4) = 3(-4)^2 + 24(-4)\]. This simplifies to: \[f(-4) = 3(16) - 96 = 48 - 96 = -48\].

Key concepts

parabola

A parabola is the graph of a quadratic function. It has a distinct shape that looks like a U or an upside-down U. The direction in which the parabola opens depends on the sign of the quadratic term's coefficient.
If the quadratic term's coefficient is positive, the parabola opens upwards. This means it will have a minimum point at its lowest part.
If the coefficient is negative, it opens downwards, leading to a maximum point at its highest part.
Understanding the shape and direction of a parabola helps in determining whether a quadratic function has a minimum or maximum value.

vertex

The vertex of a parabola is its highest or lowest point, depending on whether it opens downwards or upwards. This is a crucial point that can be found using the formula \( x = -\frac{b}{2a} \).
In the quadratic function \ (f(x) = 3x^2 + 24x) \, the coefficients are \ (a = 3) \ and \ (b = 24) \. Using these values, substitute into the formula to find the x-coordinate of the vertex: \( x = -\frac{24}{2 \times 3} = -4 \).
We then use this x-value to find the corresponding y-coordinate by plugging it back into the original function. Thus, the vertex in this exercise is at \( (-4, -48) \).

minimum value

For a parabola that opens upwards, the vertex is the point with the minimum value. In the given quadratic function \( f(x) = 3x^2 + 24x \), the minimum value occurs at the vertex.
If the quadratic coefficient (\