f(x) is decreasing when
f’ < 0
In calculus, a function f(x) is decreasing if its values are decreasing as x increases. This means that if we consider any two points on the function, as we move from left to right along the x-axis, the y-coordinate of the second point will be less than the y-coordinate of the first point.
One way to determine if a function is decreasing is to find the derivative of the function and examine its sign. If the derivative is negative, then the function is decreasing. Another way is to compare the values of the function at two different points on the interval of interest. If the value of f(x) at the second point is less than the value of f(x) at the first point, then f(x) is decreasing on the interval between those two points.
In general, a function is decreasing when:
– Its derivative is negative: f'(x) < 0 - Its second derivative is negative: f''(x) < 0 - Its slope is negative: m < 0, where m is the slope of the tangent line to the curve at any given point on the curve - Its values decrease as x increases: f(x2) < f(x1), where x2 > x1
For example, the function f(x) = -x^2 is decreasing for all values of x because its derivative is f'(x) = -2x, which is negative for all values of x. Similarly, the function f(x) = e^(-x) is decreasing for all values of x because its derivative is f'(x) = -e^(-x), which is negative for all values of x.
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