Question

Find the curve such that at each point on it the sum of the intercepts on the \(x\) and \(y\)-axes of the tangent to the curve (taking account of sign) is equal to 1 .

Short answer

The curve is given by the equation \( y = 1 - x \).

Step-by-step solution

Define the Problem Mathematically

Let the equation of the curve be given by a function, say, \( y = f(x) \). The intercepts made by the tangent at any point on the curve on the \(x\)- and \(y\)-axes should sum to 1.

Determine the Equation of Tangent

At any point \((x, y)\) on the curve, the tangent can be written as: \[ y - f(a) = f'(a)(x - a) \] where \( a \) is the point of tangency and \( f'(a) \) is the derivative.

Calculate Intercepts on Axes

The \(y\)-intercept (where \(x = 0\)) is \( y = f(a) - af'(a) \). The \(x\)-intercept (where \(y = 0\)) is \( x = a - \frac{f(a)}{f'(a)} \).

Sum the Intercepts

Sum the intercepts on the \(x\)- and \(y\)-axes: \[ (a - \frac{f(a)}{f'(a)}) + (f(a) - af'(a)) = 1 \].

Simplify the Equation

Rewrite and simplify the sum: \[ a - \frac{f(a)}{f'(a)} + f(a) - af'(a) = 1 \]. Simplifying further, obtain: \[ a(1 - f'(a)) + f(a)(1 + \frac{1}{f'(a)}) = 1 \].

Determine Differential Equation

Solve for \( f'(a) \): \[ f(a) = 1 \text{ (not dependent on \( a \))} \]. This suggests that the function is a constant function.

Solve for the Function

Since sum of intercepts is 1, the function \( f(x) \) should satisfy \[ y = 1 - x \]. Hence, the equation of the curve is: \( y = 1 - x \).

Key concepts

Tangent Line Intercepts

Understanding tangent line intercepts is essential for this problem. A tangent line to a curve at any point is a straight line that just touches the curve at that point. This line can be extended to meet the x-axis and y-axis at points called intercepts.
At any point \(a, f(a)\) on the curve, the tangent line equation can be written as: \[ y - f(a) = f'(a)(x - a) \] where \(f'(a)\) is the slope of the tangent (i.e., the derivative of \(f(x)\) at \(a\)).
The intercepts are then:

• y-intercept: found by setting \(x = 0\) in the tangent line equation, yielding \ y = f(a) - af'(a) \
• x-intercept: found by setting \(y = 0\), resulting in \ x = a - \frac{f(a)}{f'(a)} \

For this problem, the sum of these intercepts is said to be equal to 1.
Simplifying the equation: \[ (a - \frac{f(a)}{f'(a)}) + (f(a) - af'(a)) = 1 \] allows us to progress in solving the problem.

Differential Equation

A differential equation involves derivatives of a function and provides a relation between the function itself and its derivatives. In our exercise, we need to find a function \(f(x)\) whose derivative behavior allows us to satisfy a specific condition: the sum of intercepts equals 1.
From the simplified intercept equation: \[ a(1 - f'(a)) + f(a)(1 + \frac{1}{f'(a)}) = 1 \]
This sets up a differential equation that we need to solve. Solving this equation is critical for determining the desired function \(f(x)\). The differential equation essentially tells us how \(f(x)\) behaves throughout its domain.
In this particular problem, it simplifies such that \(f(a) = 1\), is not dependent on \(a\), indicating that \(f(x)\) is just a constant value, which leads us to investigate further into constant functions.

Constant Function

A constant function is a type of function where the output value remains the same, regardless of the input. Mathematically, this can be represented as \ f(x) = C \, where \(C\) is a constant.
Given the solution from our differential equation, we see that \( f(a) = 1 \) is a constant function. Here, the summation condition of the intercepts can perfectly fit a linear function where \ y = 1 - x \.
What's unique about constant functions is that their derivatives are always zero. In the context of our tangent line, using the constant function means the slope (i.e., derivative) at any point on the curve is zero.

• This confirms that the linear relationship in our problem is constant.
• Each y-value is essentially shifted by the constant intercept condition of 1 unit.

The problem leads us to understand that the function \( y = 1 - x \) is a straightforward linear equation derived from understanding intercepts and differential constraints.