Let f(x)={ x^3-x^2+10 x-5, x ≤ 1 -2 x+log2(b^2-2), x>1 . The set of values of b for wh

step 1 So in this case we need to find set of value for the f x has greater value of x equals to 1. So

Let f(x)={ x^3-x^2+10 x-5, x ≤ 1 -2 x+log2(b^2-2),...

Let $f(x)=\left\{\begin{array}{l}x^{3}-x^{2}+10 x-5, x \leq 1 \\ -2 x+\log _{2}\left(b^{2}-2\right), x>1\end{array}\right.$
The set of values of $b$ for which $f(x)$ has greatest value of $x=1$ is given by

Key Concepts

Global Maximum and Endpoints

In optimization problems, particularly with functions defined on closed or semi-closed intervals, global extrema may occur at the endpoints of the intervals. When the overall domain is split into pieces, evaluating the function at the boundary or endpoint values is necessary to decide where the maximum or minimum values occur.

Inequalities and Parameter Constraints

Determining the conditions under which a specific point is a maximum often involves setting up and solving inequalities. These inequalities ensure that the function’s value at the point in question surpasses or equals the values in its neighboring regions, thereby providing necessary restrictions on any parameters involved.

Domain Restrictions of Logarithmic Functions

Logarithmic functions require that their arguments be strictly positive. This domain restriction imposes necessary conditions on any parameters within the argument. These constraints must be taken into account when determining the allowable values of parameters so that the function remains well-defined.

Piecewise Functions

A piecewise function is defined by different expressions over different subdomains. When analyzing such a function, it is important to examine each subfunction separately, especially at the points where the definition changes, as these transition points can play a critical role in determining properties like extrema.

Continuity at a Boundary Point

Even if a piecewise function is not fully continuous, comparing the limits from each side at a boundary can help determine if a particular point might serve as a global maximum or minimum. The behavior of the function just to the left and right of the boundary contributes to the overall optimality condition.

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