If f(x)=-(1)/(3) x^3+t^2 x, where t is a real parameter. Let...

If $f(x)=-\frac{1}{3} x^{3}+t^{2} x$, where $t$ is a real parameter. Let $m(t)$ denote the minimum of $\mathrm{f}(\mathrm{x})$ over $[0,1]$ then (A) $m(t)=0$ if $t^{2} \geq \frac{1}{3}$ (B) $\mathrm{m}(\mathrm{t})=0$ for all $\mathrm{t}$ (C) $m(t)=t^{2}-\frac{1}{3}$ if $t^{2}<\frac{1}{3}$ (D) $\mathrm{m}(\mathrm{t})=\frac{1}{3}-\mathrm{t}^{2}$ for all $\mathrm{t}$.

If $f(x)=-\frac{1}{3} x^{3}+t^{2} x$, where $t$ is a real parameter. Let $m(t)$ denote the minimum of $\mathrm{f}(\mathrm{x})$ over $[0,1]$ then
(A) $m(t)=0$ if $t^{2} \geq \frac{1}{3}$
(B) $\mathrm{m}(\mathrm{t})=0$ for all $\mathrm{t}$
(C) $m(t)=t^{2}-\frac{1}{3}$ if $t^{2}<\frac{1}{3}$
(D) $\mathrm{m}(\mathrm{t})=\frac{1}{3}-\mathrm{t}^{2}$ for all $\mathrm{t}$.

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Key Concepts

Optimization over a Closed Interval

This concept involves finding the minimum or maximum value of a function when its domain is limited to a closed interval. The procedure typically includes evaluating the function at its critical points—where the derivative equals zero or is undefined—and at the endpoints of the interval. This approach ensures that the global minimum or maximum is correctly identified, as guaranteed by the Extreme Value Theorem.

Critical Point Analysis

Critical point analysis is a standard method in calculus used to find where the function's derivative is zero or fails to exist. These points are candidates for local minima or maxima. In optimization problems, especially those with parameters, identifying critical points helps determine whether the extremum occurs in the interior of the interval or is potentially achieved at an endpoint.

Parameter Dependence in Optimization

When a function includes a parameter, the location and value of its extrema can vary depending on the parameter’s value. Analyzing this dependency requires considering different cases or regimes for the parameter. This often leads to a piecewise description of the function’s minimum or maximum, with conditions that delineate which part of the domain or which candidate point gives the optimal value.

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