Let's analyze each statement to determine which one is true:
-
When k is positive, x^3 = k has one positive solution and one negative solution.
- True. The function x^3 is monotonically increasing. Therefore, for k > 0, there is one real root (solution) which is positive. There is no negative solution since x^3 = k does not cross below the x-axis for positive k.
-
When k is positive, x^3 = k has one positive solution and one negative solution.
- This is a repeat of the first statement, so it is also true.
-
When k is negative, x^2 = k has one solution, which is negative.
- False. x^2 cannot equal a negative number, as squares of real numbers are always non-negative.
-
When k is negative, x^2 = k has one solution, which is negative.
- False. The same reasoning applies. There are no real solutions because a square cannot be negative.
-
When k is positive, x^2 = k has one solution, which is positive.
- False. x^2 = k (k > 0) has two solutions: one positive (sqrt(k)) and one negative (-sqrt(k)).
-
When k is positive, x^2 = k has one solution, which is positive.
- This is a repeat of the fifth statement, and it is false for the same reason.
-
When k is negative, x^3 = k has one solution, which is negative.
- True. The function x^3 is again monotonically increasing, and for a negative k, there will be one negative solution since the function will cross the x-axis.
Based on this evaluation, the true statements are:
- When k is positive, x^3 = k has one positive solution and one negative solution.
- When k is negative, x^3 = k has one solution, which is negative.
However, since the question states to recognize "which of the following statements is true" and only allows for one selection, if you must select only one, choose either of the two lines addressing positive k and x^3. If allowed multiple answers, indicate both.