​$$\text{Let } f : \mathbb{R}^2 \to \mathbb{R} \text{ be continuous and} \\[10pt]f(t, x) < 0 \quad \text{if } tx > 0, \\[10pt]f(t, x) > 0 \quad \text{if } tx < 0. \\[10pt]\text{Consider the problem of solving the following:} \\[10pt]\dot{x} = f(t, x), \quad x(0) = 0. \\[10pt]\text{Which of the following is true?}$$​​

Correct option is A

The given differential equation

x' = f(t, x), x(0) = 0

has f(t, x) as a continuous function, and the conditions

f(t, x) < 0 for tx > 0 andf(t, x) > 0 for tx < 0

imply that f(t, x) respects the sign of tx.

This propertyensures that f(t, x) satisfies the necessary conditions for both existence anduniqueness.

By the **Picard-Lindel¨of theorem**:

since f(t, x) is continuous and satisfiesthe local Lipschitz condition in x, the differential equation has a unique localsolution.

Correct Answer: (A).