The minimum value of the expression `[3x^(2)+4x+5]` (where [.] greatest integer function) is ____________

Answer

To find the minimum value of the expression \(3x^2 + 4x + 5\) using the greatest integer function, we will follow these steps:

### Step 1: Identify the quadratic function
The given expression is a quadratic function:
\[ f(x) = 3x^2 + 4x + 5 \]

### Step 2: Determine the coefficients
In the quadratic function \(ax^2 + bx + c\):
- \(a = 3\)
- \(b = 4\)
- \(c = 5\)

### Step 3: Calculate the discriminant
The discriminant \(D\) is calculated using the formula:
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = 4^2 - 4 \cdot 3 \cdot 5 = 16 - 60 = -44 \]

### Step 4: Analyze the discriminant
Since the discriminant \(D < 0\), it indicates that the quadratic function does not intersect the x-axis and opens upwards (as \(a > 0\)). Therefore, the function has a minimum value.

### Step 5: Find the vertex
The x-coordinate of the vertex (which gives the minimum value) is found using the formula:
\[ x = -\frac{b}{2a} \]
Substituting the values:
\[ x = -\frac{4}{2 \cdot 3} = -\frac{4}{6} = -\frac{2}{3} \]

### Step 6: Calculate the minimum value
Now, substitute \(x = -\frac{2}{3}\) back into the function to find the minimum value:
\[ f\left(-\frac{2}{3}\right) = 3\left(-\frac{2}{3}\right)^2 + 4\left(-\frac{2}{3}\right) + 5 \]
Calculating each term:
\[ = 3 \cdot \frac{4}{9} - \frac{8}{3} + 5 \]
\[ = \frac{12}{9} - \frac{8}{3} + 5 \]
Converting \(-\frac{8}{3}\) to a fraction with a denominator of 9:
\[ -\frac{8}{3} = -\frac{24}{9} \]
Now, substituting:
\[ = \frac{12}{9} - \frac{24}{9} + \frac{45}{9} \]
Combining the fractions:
\[ = \frac{12 - 24 + 45}{9} = \frac{33}{9} = \frac{11}{3} \approx 3.6667 \]

### Step 7: Apply the greatest integer function
The greatest integer function (denoted by \([.]\)) takes the largest integer less than or equal to the value. Thus:
\[ \left\lfloor \frac{11}{3} \right\rfloor = 3 \]

### Final Answer
The minimum value of the expression \([3x^2 + 4x + 5]\) is:
\[ \boxed{3} \]