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The derivative f′(x)f'(x)f′(x) represents the slope of f(x)f(x)f(x).
f′(x)>0f'(x) > 0f′(x)>0 indicates f(x)f(x)f(x) is increasing.
f′(x)<0f'(x) < 0f′(x)<0 indicates f(x)f(x)f(x) is decreasing.
At a stationary point, f′(x)=0f'(x) = 0f′(x)=0.
f′(x)f'(x)f′(x) crosses xxx-axis from positive to negative.
f′(x)f'(x)f′(x) crosses xxx-axis from negative to positive.
f′(x)f'(x)f′(x) is increasing if f(x)f(x)f(x) is concave up.
f′(x)f'(x)f′(x) is decreasing if f(x)f(x)f(x) is concave down.
f′(x)f'(x)f′(x) crosses xxx-axis at each stationary point of f(x)f(x)f(x).
Look for sign changes and smoothness in the curve.
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