If f(x) is increasing, then f'(x) is?
f”(x) = positive
If f(x) is an increasing function, then it means that as x increases, the corresponding values of f(x) also increase. In other words, the slope of the tangent at any point on the graph of the function f(x) is positive.
Now, the derivative of a function f(x) is defined as f'(x) = lim(h->0) [f(x+h) – f(x)]/h, where h is a small change in x.
If f(x) is increasing, it means that as x increases, the value of f(x) also increases. This implies that f(x+h) > f(x) for any small positive value of h. Therefore, the numerator in the definition of f'(x) is positive for every h > 0.
As h approaches zero, f(x+h) gets closer and closer to f(x), meaning that the numerator approaches a positive value. Since the denominator is positive (since h > 0), it follows that the limit of the quotient, which is f'(x), is positive as well.
Therefore, we can conclude that if f(x) is an increasing function, then f'(x) is positive.
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