Find the maximum and minimum value of the quadratic function g(x) = 2x^2 + 6x + 3

Well, well, well, let me take a clownish leap into this question. To find the maximum or minimum value of a quadratic function, we can use a little trick called completing the square.

First, let's rewrite the quadratic function in vertex form. We have g(x) = 2x^2 + 6x + 3. To complete the square, we need to focus on the quadratic part, which is the 2x^2 + 6x part.

We divide everything by the leading coefficient (2) to get: g(x) = x^2 + 3x + 3/2.

Now, let's focus on the square term, which is x^2 + 3x. To complete the square, we need to add the square of half the coefficient of x. The coefficient of x is 3, so half of that would be 3/2. We square 3/2 to get 9/4.

So, we can rewrite g(x) as follows:

g(x) = (x^2 + 3x + 9/4) - 9/4 + 3/2.

Now, we can rewrite g(x) again as:

g(x) = (x + 3/2)^2 - 9/4 + 3/2.

At this point, we can observe that the square term (x + 3/2)^2 is always greater than or equal to zero. Hence, the minimum value of g(x) occurs when (x + 3/2)^2 = 0, which means x = -3/2.

To find the actual minimum value, we substitute x = -3/2 into g(x):

g(-3/2) = (-3/2 + 3/2)^2 - 9/4 + 3/2.

After some clownish calculations, we find g(-3/2) = -13/4.

So, the minimum value of g(x) is -13/4.

As for the maximum value, quadratic functions don't have a predefined maximum value. They either have a minimum value (when the coefficient of the squared term is positive) or no minimum value (when the coefficient of the squared term is negative).

Hope that brings a smile to your face!

To find the maximum and minimum values of the quadratic function g(x) = 2x^2 + 6x + 3, we need to determine whether it is a concave up or concave down function by looking at the coefficient of x^2.

Step 1: Determine the coefficient of x^2
The coefficient of x^2 is 2. Since the coefficient is positive, the function is a concave up quadratic.

Step 2: Find the vertex of the parabola
The vertex of a concave up quadratic is the minimum point.

The x-coordinate of the vertex can be found by using the formula -b/2a, where a is the coefficient of x^2 and b is the coefficient of x.

x = -6 / (2 * 2)
x = -6 / 4
x = -3/2

To find the y-coordinate of the vertex, we substitute the x-coordinate (-3/2) into the function g(x).

g(-3/2) = 2(-3/2)^2 + 6(-3/2) + 3
g(-3/2) = 2(9/4) - 9 + 3
g(-3/2) = 18/4 - 9 + 3
g(-3/2) = 9/2 - 9/1 + 3
g(-3/2) = 9/2 - 18/2 + 6/2
g(-3/2) = -3/2

The vertex is at (-3/2, -3/2) or (-1.5, -1.5).

Step 3: Determine the maximum or minimum value
Since the quadratic is a concave up function, the vertex is the minimum value of the function.

Therefore, the minimum value of the function g(x) = 2x^2 + 6x + 3 is -3/2.

As for the maximum value, since the quadratic is a concave up function, there is no maximum value. The function will continue to increase indefinitely.

In summary, the minimum value is -3/2 and there is no maximum value.

To find the maximum and minimum values of a quadratic function, we need to determine the vertex of the parabola.

The general formula for a quadratic function in the form of g(x) = ax^2 + bx + c is given by: x = -b / (2a)

In this case, the quadratic function is g(x) = 2x^2 + 6x + 3.

First, let's find the x-coordinate of the vertex using the formula x = -b / (2a):

a = 2, b = 6
x = -6 / (2 * 2)
x = -6 / 4
x = -3/2 or -1.5

The x-coordinate of the vertex is -1.5.

To find the y-coordinate of the vertex, substitute this x-coordinate back into the original function:

g(-1.5) = 2(-1.5)^2 + 6(-1.5) + 3
g(-1.5) = 2(2.25) - 9 + 3
g(-1.5) = 4.5 - 9 + 3
g(-1.5) = -1.5

The y-coordinate of the vertex is -1.5.

Therefore, the vertex is (-1.5, -1.5).

Since the coefficient of the x^2 term is positive (2 > 0), the parabola opens upwards and the vertex represents the minimum point on the graph.

Hence, the minimum value of the quadratic function g(x) = 2x^2 + 6x + 3 is -1.5.

Since the parabola opens upwards and there is no other point below the vertex, there is no maximum value for this quadratic function.

Since it is a parabola which opens upward, there is no maximum value.

g(x) = 2(x + 3/2)^2 - 3/2
so its minimum is -3/2