Question

Which of the following statements must be false about f(x)?

A) f(0) = 0
B) f(2) = 1
C) f(5) = 5
D) f(10) = 75

Answer 1

The function f(x) has a positive value and a positive slope that decreases in magnitude with increasing x. The only provided function that matches this description is y = x², implying that for f(x), f(10) should equal 100, not 75. Thus, statement D) f(10) = 75 must be false.

Step-by-step explanation:

We need to determine which statement must be false about the function f(x). Given that f(x) has a positive value at x = 3 and a positive slope that is decreasing in magnitude, let's examine the options:

1. y = 13x - This is a straight line with a constant positive slope of 13, which doesn't fit the description of a decreasing magnitude of slope.
2. y = x² - This is a parabolic function where the slope (the derivative) is 2x, which decreases as x approaches zero from the positive side and increases as x moves away from zero. This means the slope at x=3 is positive and decreases as x increases, matching the description given.

Therefore, among the options, the one that could correspond to f(x) is y = x². Hence, the statement D) f(10) = 75 must be false because if f(x) = x², then f(10) would be 100, not 75.