If the parabola y=f(x), having axis parallel to the y-axis,...

If the parabola $y=f(x)$, having axis parallel to the $y$-axis, touches the line $y=x$ at $(1,1)$, then
(A) $2 f^{\prime}(0)+f(0)=1$
(B) $2 f(0)+f^{\prime}(0)=1$
(C) $2 f(0)-f^{\prime}(0)=1$
(D) $2 f^{\prime}(0)-f(0)=1$

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves expressing functions in standard form, evaluating them at particular points, and solving resulting equations to find relationships between coefficients. In the context of quadratic functions and tangency problems, it helps in deducing conditions that the parameters must satisfy by using both the function value at the tangency point and the matching derivative condition.

Derivative

The derivative of a function quantifies the rate at which the function's value changes with respect to changes in its input, which geometrically represents the slope of the tangent to the curve at any given point. In problems involving tangency, equating the derivative of the curve to the slope of the line at the point of contact is essential in establishing relationships between the function's parameters.

Parabola

A parabola is the graph of a quadratic function, usually written in the form y = ax² + bx + c, where a, b, and c are constants. Its axis of symmetry is vertical when the quadratic term’s coefficient is nonzero. Understanding the standard form and properties of a parabola is fundamental when analyzing its geometric characteristics and behavior, particularly in relation to other curves or lines.

Tangency Condition

When a curve touches a line exactly at one point without crossing it, the line is said to be tangent to the curve. This condition requires that at the point of tangency, both the function values are equal and the derivatives (or slopes) are identical. Applying these conditions provides key constraints that help determine the parameters of a quadratic function when a line touches its graph.

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