If f'(x)<0, f(x) is
If f'(x) is negative, it means that the derivative of the function f(x) with respect to x is negative
If f'(x) is negative, it means that the derivative of the function f(x) with respect to x is negative. In mathematical terms, this means that the function f(x) is decreasing.
To understand this concept, let’s recall the definition of the derivative. The derivative of a function f(x) at a specific point x measures the rate at which the value of the function is changing at that point. If the derivative is negative, it means that as we move along the x-axis in the positive direction, the value of the function is decreasing.
In other words, if the derivative f'(x) is negative, f(x) is decreasing as x increases.
For example, let’s say we have a function f(x) = x^2. The derivative of this function with respect to x is f'(x) = 2x. Since the value of 2x is always positive for values of x greater than zero, the function f(x) = x^2 is always increasing.
On the other hand, if we have a function f(x) = -x^2, the derivative is f'(x) = -2x. Here, the value of -2x is always negative for values of x greater than zero, which means that the function f(x) = -x^2 is always decreasing.
So, if f'(x) is negative, it implies that the function f(x) is decreasing.
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