Problem 71 - Calculus Single Variable

Question

Suppose \(a \neq 0 .\) What relationship between \(a\) and \(b\) is a necessary and sufficient condition for the graph of \(f(x)=a x^{2}+b\) to have a tangent line that passes through the origin?

Step-by-Step Solution

Answer

The necessary and sufficient condition is \(b = ax_0^2\) for some \(x_0\).

Step 1: Define Tangent Line

In order for the tangent line to a curve at some point \((x_0, f(x_0))\) to pass through the origin \((0, 0)\), the line must satisfy the equation \(y = mx\), where \(m\) is both the slope of the tangent line and equal to the derivative of the function at \(x_0\).

Step 2: Differentiate the Function

Find the derivative of the function \(f(x) = ax^2 + b\). The derivative \(f'(x)\) represents the slope of the tangent line to the curve at any point \(x\).The derivative is \(f'(x) = 2ax\).

Step 3: Find the Point of Tangency

For the tangent line to pass through the origin, the point \((x_0, f(x_0))\) must also satisfy \(f(x_0) = m x_0\), where \(m = f'(x_0)\). Thus, we have \(f(x_0) = f'(x_0) x_0\).Substituting \(f'(x_0) = 2ax_0\), we need \(ax_0^2 + b = 2ax_0^2\).

Step 4: Solve for the Condition on \(b\)

To satisfy the tangent line passing through the origin, we equate \(ax_0^2 + b = 2ax_0^2\) to simplify for \(b\):Subtract \(ax_0^2\) from both sides, obtaining \(b = ax_0^2\).

Step 5: Necessary and Sufficient Condition

The necessary and sufficient condition for the graph of \(f(x)\) to have a tangent line that passes through the origin is \(b = ax_0^2\) for some \(x_0\).

Key Concepts

Tangent line

To understand the concept of a tangent line, think of it as a line that just "touches" a curve at a specific point. This line represents the instantaneous direction in which the curve is heading at that point. One crucial property of a tangent line is that it shares the same slope as the curve at their point of contact. When you're trying to find a tangent line that passes through the origin (the point where both x and y are zero), you'll set the line equation as \(y = mx\). Here, \(m\) is the slope of the tangent line at a specific point on the curve. To ensure this line also intersects the origin, the curve must pass through the origin when evaluated at that point. Hence, a tangent line can be characterized by both its slope and the specific curve point of contact.

Quadratic functions

Quadratic functions are a type of polynomial function given by a formula of the form \(f(x) = ax^2 + bx + c\). In this case, the function is \(f(x) = ax^2 + b\), where \(a\) and \(b\) are real numbers, and \(c\) is zero. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \(a\).
Quadratic functions have key features that include the vertex, the axis of symmetry, and possibly x-intercepts and a y-intercept. The vertex can be a maximum or minimum point depending on whether the parabola opens upwards or downwards. Understanding these functions helps when determining where a tangent line can exist for a given quadratic.

Necessary and sufficient conditions

A "necessary and sufficient" condition is a condition that must be true for a certain statement to be true and, by itself, is enough to ensure the statement's truth. To illustrate this with the problem at hand, consider the statement: the graph of \(f(x) = ax^2 + b\) has a tangent line that passes through the origin if, and only if, \(b = ax_0^2\) for some \(x_0\).
This mathematical language implies two things:

• "Necessary" means if the graph can have such a tangent, then \(b = ax_0^2\) must hold true.
• "Sufficient" means if \(b = ax_0^2\) holds, the graph will definitely have a tangent passing through the origin.

So, the condition acts as both a requirement and a guarantee, providing a complete description of when the tangent line exists as specified.