The set of all values of the parameters a for which the...

The set of all values of the parameters $a$ for which the points of minimum of the function $y=1+a^{2} x-$ $x^{3}$ satisfy the inequality $\frac{x^{2}+x+2}{x^{2}+5 x+6} \leq 0$ is (a) an empty set (b) $(-3 \sqrt{3},-2 \sqrt{3})$ (c) $(2 \sqrt{3}, 3 \sqrt{3})$ (d) $(-3 \sqrt{3},-2 \sqrt{3}) \cup(2 \sqrt{3}, 3 \sqrt{3})$

The set of all values of the parameters $a$ for which the points of minimum of the function $y=1+a^{2} x-$ $x^{3}$ satisfy the inequality $\frac{x^{2}+x+2}{x^{2}+5 x+6} \leq 0$ is
(a) an empty set
(b) $(-3 \sqrt{3},-2 \sqrt{3})$
(c) $(2 \sqrt{3}, 3 \sqrt{3})$
(d) $(-3 \sqrt{3},-2 \sqrt{3}) \cup(2 \sqrt{3}, 3 \sqrt{3})$

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Key Concepts

Parameter Dependent Functions

Parameter dependent functions include parameters that affect the behavior of the function. Analyzing how the location and nature of extrema depend on these parameters is crucial in understanding the full behavior of the function under varying conditions.

Rational Inequalities

Rational inequalities consist of ratios of polynomials that are compared to zero. Solving these inequalities generally requires factoring the polynomials, determining the zeros of the numerator and the denominator, and then performing a sign analysis over the intervals defined by these zeros to identify where the inequality holds.

Optimization of Functions

Optimization involves finding the extreme values—either minima or maxima—of a function. This is typically done by taking the derivative, setting it to zero to locate critical points, and then using tests to confirm whether these points are minima, maxima, or saddle points.

Critical Points

Critical points are values at which the derivative of a function is zero or undefined. They indicate where the function's rate of change shifts, and are essential in determining local minima and maxima in optimization problems.

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