If you know that a function f(x) is positive everywhere, what can you conclude from that about the derivative?

d) You can cook up a function which does a, or b, or c, or any combination of those properties. Take y = e^x. That derivative is positive and increasing everywhere. Take y = e^(-x). That derivative is negative and increasing everywhere (toward 0). Take y = 1 - e^(-x). That derivative is positive and decreasing everywhere (toward 0). Take y = sin(x) + 3. That's positive and the derivative is oscillating, having none of the above properties.

Let's run through the options: If the derivative is positive that means the function is increasing everywhere, but this doesn't have to be the case. Consider f(x) = 1/(1 + x²). f'(x) < 0 for all x > 0. The same example shows that the derivative doesn't have to be increasing everywhere: f'(x) decreases about around. Again the same example shows that the derivative doesn't have to be concave upward because f'(x) is concave down around (-1/2, 0) and a positive interval. You can tell I didn't want to explicitly calculate all the intervals because it would require the third derivative, but this example is good for disproving all but the last choice. Just always keep in mind that the derivative speaks to the shape of a function, not its value.

d) You can cook up a function which does a, or b, or c, or any combination of those properties. Take y = e^x. That derivative is positive and increasing everywhere. Take y = e^(-x). That derivative is negative and increasing everywhere (toward 0). Take y = 1 - e^(-x). That derivative is positive and decreasing everywhere (toward 0). Take y = sin(x) + 3. That's positive and the derivative is oscillating, having none of the above properties.