Let f(x)=a x^2+b x+c, where a>0 . Prove that f(x) ≥0 for all x if and only if b^2-4 a c ≤0 . [Hi

step 1 In the question, let Fx be equal to AX squared plus BX plus C, where A is greater than 0, prove

Let f(x)=a x^2+b x+c, where a>0 . Prove that f(x) ≥0 for all...

Key Concepts

Quadratic Function

A quadratic function is a polynomial of degree two typically written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ? 0. This function models a parabola in the Cartesian plane, and its properties, including its curvature and orientation, are governed by its coefficients.

Parabola and its Vertex

The graph of a quadratic function is a parabola. When a > 0, the parabola opens upward, meaning that it has a minimum point. The vertex of the parabola represents this minimum. Finding the vertex by methods such as completing the square or using the formula x = -b/(2a) allows one to determine the minimum value of the quadratic function.

Discriminant

The discriminant of a quadratic function is given by the expression b² - 4ac. It provides critical information about the number and nature of the roots (or x-intercepts) of the quadratic equation. A negative discriminant indicates that there are no real roots, while a zero discriminant indicates a repeated real root, and a positive discriminant indicates two distinct real roots.

Nonnegativity Condition for Quadratics

For a quadratic function that opens upward (a > 0), the function will be nonnegative for all real values of x if its minimum value is at least zero. This condition is directly connected to the discriminant, as a non-positive discriminant (b² - 4ac ? 0) guarantees that the quadratic has no real roots (or exactly one, in the case of zero discriminant) and thus does not cross the x-axis, ensuring that the function is always greater than or equal to zero.

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