Let f(x)=x^3-3(7-a) x^2-3(9-a^2) x+2 The values of a, if f(x) has a positive point of local maxi

step 1 So, in this question we have when, when point of maximum is positive, positive point of minima i

Let f(x)=x^3-3(7-a) x^2-3(9-a^2) x+2 The values of a, if...

Let $f(x)=x^{3}-3(7-a) x^{2}-3\left(9-a^{2}\right) x+2$
The values of $\mathrm{a}$, if $\mathrm{f}(\mathrm{x})$ has a positive point of local maxima, are
(A) $\phi$
(B) $(-\infty,-3) \cup(3, \infty)$
(C) $\left(-\infty, \frac{58}{14}\right)$
(D) None of these

Key Concepts

First and Second Derivative Tests

The first derivative test involves analyzing the sign changes of a function’s derivative around critical points to classify them as local maxima or minima. The second derivative test uses the concavity of the function—if the second derivative at a critical point is negative, it indicates a local maximum; if positive, a local minimum.

Local Extrema

Local extrema refer to points on the graph of a function where the function value is either higher (local maximum) or lower (local minimum) compared to nearby points. Determining whether a critical point is a local maximum or minimum often involves using derivative tests.

Parameter Dependence in Function Analysis

When functions involve parameters, the behavior and characteristics of the function, such as the location and nature of local extrema, can vary based on the parameter values. Analyzing how parameters affect the function’s derivatives and critical points is essential for understanding the overall behavior of parameterized functions.

Cubic Polynomial Functions

Cubic functions are third-degree polynomials characterized by their general form and the possibility of having one or two turning points. Their diverse behavior, such as inflection points and the potential for one local maximum and one local minimum, makes them critical in studying function behavior and curve sketching.

Critical Points

Critical points are values in the domain of a function where its derivative is zero or undefined, indicating potential locations for local extrema (maxima or minima). Finding these points is a fundamental step in analyzing the behavior and shape of any differentiable function.

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