f(x) is increasing -> f'(x) is
positive
If f(x) is increasing, it means that as x increases, so does the value of f(x). Therefore, the derivative (f'(x)) of f(x) must also be positive. This is because the derivative represents the slope of the curve at any given point. An increasing function means that the slope of the curve is positive, which is reflected by the positive value of the derivative.
Mathematically, we can express this as:
f(x) is increasing if f(x+h) > f(x) for any h > 0.
To find f'(x), we can apply the limit definition of the derivative:
f'(x) = lim(h->0) [(f(x+h) – f(x))/h]
Since we know that f(x+h) > f(x), it follows that f'(x) must be positive. If we take the limit as h approaches 0, we get the slope (derivative) of the curve at x. If the slope is positive, the function is increasing.
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