Assume that f^'(x) is continuous everywhere and that f(x)...

Assume that $f^{\prime}(x)$ is continuous everywhere and that $f(x)$ has one and only one critical value at $x=0$. Use the additional given information to determine whether $y=f(x)$ attains a relative minimum, a relative maximum, or neither at $x=0 .$ Explain your reasoning. Sketch a possible graph in each case. $f(0)=1, f(5)=5,$ and $f(-5)=0$

Key Concepts

Critical Point

A critical point of a function is a point in its domain where the derivative is zero or undefined. These points are potential candidates for local maxima, minima, or points of inflection. Understanding the definition of a critical point is essential for analyzing a function's behavior because it indicates where changes in increasing or decreasing trends may occur.

Relative Extrema

Relative extrema refer to points where a function attains a local maximum or minimum value compared to nearby points. A local maximum is a point where the function value is greater than that at immediately adjacent points, whereas a local minimum is where it is lower. Identifying relative extrema helps in understanding the overall shape and behavior of the function.

First Derivative Test

The first derivative test involves analyzing the sign of the derivative before and after a critical point to determine whether that point is a local maximum, a local minimum, or neither. If the derivative changes its sign (from positive to negative or vice versa) around the critical point, the function has a local extremum; if not, the function may simply have a horizontal tangent, often indicating a point of inflection.

Point of Inflection

A point of inflection is a point on the graph of a function where the curvature changes, which typically means the function changes concavity. When the derivative is zero at an inflection point but does not correspond to a relative extremum, it indicates that the function’s rate of increase or decrease does not change in an 'extremal' way. Recognizing inflection points is important for a complete understanding of the function's graph.

Continuity of the Derivative

The fact that a function’s derivative is continuous everywhere ensures that the behavior of the derivative, including its sign changes and zeros, is well-behaved. This property is important when applying the first derivative test because it guarantees there are no sudden jumps or discontinuities that might complicate the analysis of the function's local extremum and inflection points.

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