Let f(x)={ -x^3+log2 b 0<x<1 3 x x ≥ 1 .. Then, the set of values of b for whi

step 1 In the question, we are given a function of x, that is minus x cube plus log base 2, and for the

Let f(x)={ -x^3+log2 b 0<x<1 3 x x ≥ 1 .....

Let $f(x)=\left\{\begin{array}{cc}-x^{3}+\log _{2} b & 0<x<1 \\ 3 x & x \geq 1\end{array}\right.$. Then, the set of
values of $b$ for which $f(x)$ has least value at $x=1$ is
(A) $R^{+}$
(B) $(0,16]$
(C) $[16, \infty)$
(D) None of these

Key Concepts

Piecewise Functions

Piecewise functions are defined by different expressions on different intervals of the domain. Understanding them involves analyzing each part of the function on its respective subdomain, especially at the boundaries where the definitions change, to evaluate properties such as continuity and extrema.

Global Extremum (Minimum) Analysis

Determining the global minimum of a function involves ensuring that a specific point yields a value that is lower than or equal to the function’s values throughout its entire domain. For piecewise functions, this can require matching the values at the boundary points and comparing behavior on either side of the transition.

One-sided Limits and Boundary Behavior

When assessing functions defined by different rules on adjacent intervals, it is crucial to examine the one-sided limits at the boundaries. This analysis helps in determining whether an extreme value, such as a minimum, occurs at the boundary by comparing the left-hand and right-hand behaviors.

Parameter Effects and Logarithmic Transformations

Parameters within a function, especially when incorporated into a logarithmic transformation, can shift the graph vertically or alter its behavior. Analyzing how changes in the parameter affect the function’s values is key to finding the range of parameter values that satisfy a specific condition, such as ensuring the minimum occurs at a given point.

Transcript

Please give Ace some feedback