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

Suppose that $f^{\prime}(x)$ is continuous everywhere and $f(x)$ has two and only two critical values. Explain why it is not possible for $f$ to have the following values: $f(0)=10$ $f(1)=5, f(2)=8, f(3)=6, f(4)=9$

Key Concepts

Critical Points

Critical points are points in the domain of a function where its derivative is zero or does not exist. In the context of this problem, the function f has a continuous derivative throughout, which means that every time the function’s behavior changes from increasing to decreasing or vice versa, a critical point with f?'(x) = 0 must occur. A function with only two critical points thus can only change its monotonic behavior (from increasing to decreasing or the reverse) at most twice.

Local Extrema

Local extrema (local minima and local maxima) occur at critical points where the function attains a value lower or higher than nearby points. The hypothesis that f has exactly two critical values typically implies that there is only one local maximum and one local minimum. If the function’s value pattern forces additional fluctuations, it would necessitate extra local extrema, which is not possible under the given conditions.

Monotonicity and Sign Changes

Monotonicity refers to intervals where a function is exclusively increasing or decreasing. A function that alternates from decreasing to increasing and then back again must have changes in monotonicity, each of which is signaled by a zero derivative. In the provided sequence of function values, there are more sign changes (i.e., more alternations between increasing and decreasing behavior) than allowed by only two critical points, leading to a contradiction.

Intermediate Value and Rolle’s Theorems

The Intermediate Value Theorem and Rolle’s Theorem are tools used to infer the existence of critical points within intervals. Rolle’s Theorem, in particular, guarantees that if a function takes on values that force it to oscillate (for example, decreasing then increasing), there must be a point where the derivative is zero. The given function values force multiple oscillations, implying more than two points where f?'(x) is zero, which contradicts the assumed limitation on the number of critical points.

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