The function \( f(x) = 2x^2 - 8x + 6 \) has a minimum value. What is this value?
["The Function ( f(x) = 2x^2 - 8x + 6 ) Achieves a Minimum Value – Here’s What It Is", "When analyzing quadratic functions, one key concept is identifying whether the function reaches a minimum or maximum value. The function ( f(x) = 2x^2 - 8x + 6 ) is a classic example of a quadratic with a parabolic shape that opens upward due to its positive leading coefficient. Because of this, the function has a minimum value at its vertex, not a maximum.", "### Understanding the Vertex of a Parabola", "For a quadratic function in the standard form:", "[\nf(x) = ax^2 + bx + c\n]", "the x-coordinate of the vertex, which determines the location of the minimum (or maximum), is given by:", "[\nx = -\frac{b}{2a}\n]", "In our function ( f(x) = 2x^2 - 8x + 6 ):\n- ( a = 2 )\n- ( b = -8 )\n- ( c = 6 )", "Plugging into the vertex formula:", "[\nx = -\frac{-8}{2 \cdot 2} = \frac{8}{4} = 2\n]", "This means the function reaches its minimum value at ( x = 2 ).", "### Calculating the Minimum Value", "To find the actual minimum value of the function, substitute ( x = 2 ) back into ( f(x) ):", "[\nf(2) = 2(2)^2 - 8(2) + 6 = 2(4) - 16 + 6 = 8 - 16 + 6 = -2\n]", "### Why This Value Matters", "Knowing this minimum value is crucial in various applications, from optimizing area and profit in economics to determining peak efficiency in engineering problems. Since the parabola opens upward (( a = 2 > 0 )), the vertex represents the lowest point on the graph — the global minimum for this function.", "### Summary", "- The function ( f(x) = 2x^2 - 8x + 6 ) has a minimum value because it opens upward.\n- The minimum occurs at ( x = 2 ).\n- The minimum value is ( f(2) = -2 ).", "Understanding the vertex and evaluating the function at this point provides clear insight into the behavior of any quadratic. Recognizing this minimum not only aids in graphing but also supports decision-making in real-world modeling scenarios.", "---", "Key Takeaway:\nThe minimum value of the quadratic function ( f(x) = 2x^2 - 8x + 6 ) is (-2), achieved when ( x = 2 ). This knowledge empowers deeper analytical and practical problem solving."]
