A function is shown.
$f(x)=5(x-2)^2+3$
What is the minimum value of the function?
The given function is \(f(x) = 5(x - 2)^2 + 3\). This is a quadratic function written in the standard vertex form, which is given by \(f(x) = a(x - h)^2 + k\). In this expression, the point \((h, k)\) represents the vertex of the parabola, and the coefficient \(a\) determines the direction in which the parabola opens. If \(a > 0\), the parabola opens upwards, meaning the vertex corresponds to the minimum value of the function. If \(a < 0\), the parabola opens downwards, and the vertex corresponds to the maximum value of the function.
By comparing our function \(f(x) = 5(x - 2)^2 + 3\) to the vertex form \(f(x) = a(x - h)^2 + k\), we can identify the parameters:
Since \(a = 5\), which is greater than \(0\), we know that the parabola opens upwards, and the function will have a global minimum value at its vertex.
The vertex of this parabola is at the point \((h, k) = (2, 3)\). For any parabola that opens upwards (\(a > 0\)), the minimum value of the function is the \(y\)-coordinate of its vertex, which is \(k\). Therefore, the minimum value of \(f(x)\) is \(3\). This confirms that as \(x\) varies, the smallest value \(f(x)\) can attain is \(3\), occurring when \(x = 2\).
Consider a company that produces a specific type of gadget. The cost of production often follows a quadratic model, where the cost \(C\) depends on the quantity produced \(x\). If the cost function is \(C(x) = 5(x - 2)^2 + 3\) (in thousands of dollars), finding the vertex helps the company determine the production level \(x\) that minimizes the total cost. In this case, the minimum cost is \(\$3,000\), which occurs when the production quantity is \(2\) units. Similar quadratic models are used in physics to find the minimum energy state of a system or in geometry to optimize areas and volumes.