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. $f(-5)=-50, f(10)=100$ and $f(0)=0$

Key Concepts

Monotonicity

Monotonicity describes the behavior of a function in terms of whether it is consistently increasing or decreasing on an interval. If a function is monotonic (only increasing or only decreasing) across an interval, any horizontal tangent (where the derivative is zero) that occurs does not correspond to a local extremum. Instead, it may represent a stationary point such as a point of inflection.

Stationary Points and Inflection Points

A stationary point is where the derivative is zero, and such a point can either correspond to a relative extremum or to a point of inflection. An inflection point is where the function changes concavity (from concave up to concave down or vice versa), but it is not necessarily an extremum. In the context of a continuous derivative and a unique critical point, if the function is monotonic on either side of the critical point, then the stationary point is not a relative extremum but rather a point of inflection.

Relative Extrema

Relative extrema refer to local minimum or maximum values of a function. A local minimum is a point where the function’s value is lower than at nearby points, while a local maximum is a point where the function’s value is higher than at nearby points. Determining whether a critical point represents a relative extremum involves checking the function’s behavior on intervals immediately to the left and right of that point.

Critical Points

A critical point is any input value in the domain of a function where the derivative is zero or undefined. They are important because they are potential candidates for local extrema (local minima or maxima) or points of inflection. In the context of this problem, the function has one and only one critical point at x = 0, which directs our focus toward that point when analyzing the function's behavior.

First Derivative Test

The first derivative test involves examining the sign of the derivative on either side of a critical point to determine the behavior of the function. If the derivative changes from negative to positive, the function has a relative minimum; if it changes from positive to negative, the function has a relative maximum. In situations where the derivative does not change sign, the critical point may be a point of inflection rather than an extremum.

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