Question

Let f and g be inverse functions that are differentiable for all x. If f'(x) = g'(x) for all x, which of the following statements must be false?

1) f(x) and g(x) have the same derivative for all x
2) f(x) and g(x) have the same constant term
3) f(x) and g(x) have the same leading coefficient
4) f(x) and g(x) have the same degree

Answer 1

The false statement among the given options is that 'f(x) and g(x) have the same derivative for all x' since the derivatives of inverse functions are reciprocals of each other, not identical. Hence, statement 1 is false.

Step-by-step explanation:

The question revolves around two differentiable inverse functions f and g, where it is given that f'(x) = g'(x) for all x. If f and g are indeed inverse functions, statement 1) stating that 'f(x) and g(x) have the same derivative for all x' must be false, because the derivatives of inverse functions are not the same; rather, they are reciprocals of each other when evaluated at corresponding points.

More formally, if f and g are inverse functions, then f(g(x)) = x. By the chain rule of differentiation, f'(g(x))g'(x) = 1, implying that f'(g(x)) = 1/g'(x) when g'(x) is not zero. This is a contradiction to the given information that f'(x) = g'(x), thus statement 1) must be the false statement.

As for statements 2), 3), and 4), they make no logical sense in the context of inverse functions, without further information these statements cannot typically be judged true or false.