The curve y=a x^2+b x+c passes through the point (1,2) and...

The curve $y=a x^{2}+b x+c$ passes through the point $(1,2)$ and its tangent at origin is the line $y=x$. The area bounded by the curve, the ordinate of the curve at minima and the tangent line is
(a) $\frac{1}{24}$
(b) $\frac{1}{12}$
(c) $\frac{1}{8}$
(d) $\frac{1}{6}$

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two and are usually given in the form y = ax² + bx + c. They form parabolic curves that open upwards if a > 0 or downwards if a < 0. Their structure allows determination of key features such as the axis of symmetry, vertex, and roots, which are essential for analyzing the behavior and intersections of the curve with lines.

Tangent Lines and Derivatives

The tangent line to a curve at a given point represents the instantaneous rate of change of the function at that point, and its slope is given by the derivative. For a quadratic function, the derivative is a linear function, and setting it equal to the slope of a proposed tangent line provides a condition that helps determine the coefficients of the quadratic.

Vertex and Extreme Values

The vertex of a quadratic function represents the minimum point if the parabola opens upward (a > 0) or the maximum point if it opens downward (a < 0). The vertex can be found using the formula x = -b/(2a), and it is crucial in optimization problems, such as finding bounded areas defined by the curve, as it often serves as a natural boundary or a point of tangency.

Area Between Curves

Finding the area between curves involves determining the intersection points of the curves and integrating the difference of the functions across the interval defined by these intersections. This technique is a fundamental application of integration in calculus and is used to calculate the exact region enclosed between a given function and a line or another function.

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