f'(x)=0 & f”(x)<0
f(x) has a maximum
A function f(x) has a critical point at x=c when f'(c)=0. Therefore, the first condition f'(x)=0 ensures that we are dealing with a critical point.
The second condition f”(x)<0 indicates that the function is concave down at that critical point. In other words, as we move from left to right along the graph of the function, the slope of the tangent line is decreasing. This means that the function reaches a maximum at that critical point, as any movement away from that point in either direction will result in a slope that is positive again. Overall, when f'(x)=0 and f''(x)<0, we can conclude that we have a critical point on the function that is a local maximum.
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