The global maximum value of f(x)=log10(4 x^3-12 x^2+11 x-3), x ∈[2,3] is (A) -(3)/(2) log10 3 (B

step 1 So the question in this question we have y is equal to 4x cube minus 12 x squared plus 11 x minu

The global maximum value of f(x)=log10(4 x^3-12 x^2+11 x-3),...

The global maximum value of $f(x)=\log _{10}\left(4 x^{3}-12 x^{2}+11 x-3\right), x \in[2,3]$ is
(A) $-\frac{3}{2} \log _{10} 3$
(B) $1+\log _{10} 3$
(C) $\log _{10} 3$
(D) $\frac{3}{2} \log _{\mathrm{t} 0} 3$

Key Concepts

Logarithm Properties

Logarithmic functions, defined as the inverses of exponential functions, are strictly increasing when the base is greater than 1. This property means that when optimizing a function that is composed with a logarithm, the location of the optimum of the inner function directly determines the optimum of the logarithmic composition, as long as the inner function remains positive.

Global Optimization on Closed Intervals

When a function is continuous on a closed interval, the Extreme Value Theorem guarantees that both a global maximum and a global minimum exist on that interval. Finding these extrema typically involves evaluating the function at the endpoints of the interval and at any critical points within the interval.

Polynomial Functions and Their Analysis

Polynomial functions are continuous and differentiable over the entire set of real numbers, which makes them amenable to analysis through calculus. Understanding their behavior, particularly by finding roots and critical points, is essential in determining where a maximum or minimum might occur.

Differentiation and Critical Points

The process of differentiation provides the derivative of a function, which is used to locate critical points by setting the derivative equal to zero or where it does not exist. These critical points, along with the interval endpoints, must be evaluated to determine the global maximum or minimum values of the function.

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