f'(x)=0 & f”(x)<0
f(x) has a maximum
When f'(x)=0 and f”(x)<0, it means that the function f is at a critical point, and that the concavity of the function is negative at this point. A critical point occurs where the slope of the function is zero, either indicating a local maximum or a local minimum. The second derivative, f''(x), indicates the concavity of the function: if f''(x)<0, the function is concave downward, which means that the tangent lines to the graph are decreasing as we move from left to right. This indicates that the critical point is a local maximum. To understand this conceptually, imagine a parabolic curve facing downward. If you imagine a vertical line drawn through the vertex of the curve, this line will intersect the parabola at a point where the slope is zero. The second derivative of the function is always negative in this case, as the concavity of the curve is downward. Thus, this point is a maximum. In summary, if f'(x)=0 and f''(x)<0, it indicates that the function has a local maximum at that point. This knowledge can be used in various applications, such as optimization problems and maximum/minimum value problems.
More Answers:
Intermediate Value Theorem: Calculus Concepts and Proving Real Roots in Functions.Continuity in Mathematics: Definition, Properties, and Examples.
Vertices in Geometry and Graph Theory: Explained with Examples