If f'(x) is decreasing, then f”(x) is?
f'(a)
If f'(x) is decreasing, it means that the rate at which the function f(x) is increasing is decreasing. In other words, the slope of the tangent line to f(x) is decreasing as x increases. This can happen if the curve is becoming flatter and flatter or if it is concave down.
Now, the second derivative, f”(x), represents the rate at which the slope of the tangent line is changing. If a function is decreasing, its slope will become less and less steep, which means that its second derivative f”(x) must be negative. In other words, if f'(x) is decreasing, then f”(x) is negative or the function is concave down.
Therefore, we can conclude that if f'(x) is decreasing, then f”(x) is negative or the function is concave down.
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