91Ó°ÊÓ

A local minimum is a point where a function has a value smaller than at any other nearby points. For a quadratic function \( f(x) = ax^2 + bx + c \), it occurs at the vertex. In our exercise, the local minimum is at \( (1, -1) \). This tells us that when \( x = 1 \), the value of the function \( f(x) \) is \(-1\). To find the local minimum, we also look at the derivative. If a function has a local minimum at a point, the derivative at that point must be zero. We use this information to help find our quadratic function.

The derivative of a function measures how the function's output changes as the input changes. For a quadratic function \( f(x) = ax^2 + bx + c \), the derivative is \( f'(x) = 2ax + b \). In our exercise, we know the local minimum occurs at \( x = 1 \). Because the slope of the tangent line at a local minimum is zero, we set \( f'(1) = 0 \). This gives us the equation \( 2a(1) + b = 0 \). We'll use this equation to help find the values of \( a \) and \( b \) needed to define our function. The concept of the derivative helps us understand how changes in \( x \) affect changes in \( f(x) \) and assists in locating key points such as local minima or maxima.

A system of equations consists of two or more equations with the same set of variables. To solve for \( a \), \( b \), and \( c \) in our quadratic function, we use the information given to create a system. From the initial conditions, we know the function passes through the points \( (0, 1) \) and has a local minimum at \( (1, -1) \). This gives us the equations: \( f(0) = c = 1 \) and \( f(1) = a(1)^2 + b(1) + c = -1 \). Adding the condition from the derivative, \( 2a + b = 0 \), we construct a system of three equations. We then solve this system step-by-step to find our coefficients \( a \), \( b \), and \( c \).

A quadratic function is a polynomial function of degree two, with the general form \( f(x) = ax^2 + bx + c \). The graph of a quadratic function is a parabola, which opens upwards if \( a > 0 \) and downwards if \( a < 0 \). In our problem, we are given specific points and a local minimum. The process involves constructing the function so it meets these criteria. We started with the equation \( c = 1 \) as the function passes through \( (0, 1) \). Then, we used the local minimum information and the derivative to form a system of equations. Solving this system, we found \( a = 1 \) and \( b = -2 \). Thus, the quadratic function that fits all conditions is \( f(x) = x^2 - 2x + 1 \). Quadratic functions are crucial in many areas, including physics, engineering, and economics, due to their properties and the way they model real-world phenomena.