Question

Let f be a differentiable function such that f'(x) > 1 for all x. If a < b, which of the following statements could be false?

1) f(a) < f(b)
2) f(a) = f(b)
3) f(a) > f(b)
4) f'(a) < f'(b)

Answer 1

If f'(x) > 1 for all x, then statement 1 is true, statement 2 is false, statement 3 is false, and statement 4 is true.

Step-by-step explanation:

If $f'(x) > 1$ for all $x$, then $f(x)$ is an increasing function. This means that if $a < b$, we can say that $f(a) < f(b)$. So, statement 1) $f(a) < f(b)$ is true.

Since $f(x)$ is an increasing function, it cannot have the same value at different points. So, statement 2) $f(a) = f(b)$ is false.

Similarly, since $f(x)$ is an increasing function, if $a < b$, it follows that $f(a) < f(b)$. So, statement 3) $f(a) > f(b)$ is false.

Since we know that $f(x)$ is an increasing function, it means that the slope of $f(x)$, denoted by $f'(x)$, is also increasing. Therefore, if $a < b$, we can conclude that $f'(a) < f'(b)$. Hence, statement 4) $f'(a) < f'(b)$ is true.