Identify the correct statements: (A) If f(x)=a x^3+b and f...

Identify the correct statements: (A) If $f(x)=a x^{3}+b$ and $f$ is strictly increasing on $(-1,1)$ then $\mathrm{a}>0$. (B) An $n$ th-degree polynomial has atmost ( $n-1)$ critical points. (C) If $\mathrm{f}^{\prime}(\mathrm{x})>0$ for all real numbers $\mathrm{x}$, then $\mathrm{f}$ increases without bound. (D) The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.

Identify the correct statements:
(A) If $f(x)=a x^{3}+b$ and $f$ is strictly increasing on $(-1,1)$ then $\mathrm{a}>0$.
(B) An $n$ th-degree polynomial has atmost ( $n-1)$ critical points.
(C) If $\mathrm{f}^{\prime}(\mathrm{x})>0$ for all real numbers $\mathrm{x}$, then $\mathrm{f}$ increases without bound.
(D) The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.

Key Concepts

Strictly Increasing Functions

This concept refers to functions that, for any two points in their domain, preserve the strict order — if x? < x? then f(x?) < f(x?). It is crucial in understanding conditions under which a function is monotonic. For example, in polynomials modified by parameters, the sign of the leading coefficient often determines whether the function is strictly increasing or decreasing across an interval.

Critical Points in Polynomials

A critical point occurs when the derivative of a function is zero or undefined. For an n??-degree polynomial, its derivative is an (n-1)??-degree polynomial, which means it can have at most n-1 distinct real roots (critical points). This concept is key when analyzing the behavior of polynomial functions and determining potential extrema.

Increasing Functions and Unboundedness

A common misconception is that if a function's derivative is positive everywhere, then the function must increase without bound. However, while a positive derivative ensures that the function is strictly increasing, it does not automatically guarantee that the function will tend to infinity, as the overall behavior depends on additional properties of the function.

Extreme Value Theorem

This theorem states that a continuous function on a closed interval will attain both a maximum and a minimum value. Importantly, the maximum (or minimum) value can be achieved at more than one point in the interval if the function is constant over a region or has a plateau, demonstrating that the location of extrema need not be unique even though the extreme values themselves are well-defined.

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