What is the minimum value of 2^{((x^2 - 3)^3 | vedclass

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Question

(C) The expression $2^{((x^2 - 3)^3 + 27)}$ attains its minimum value when the exponent $((x^2 - 3)^3 + 27)$ is minimized.Let $f(x) = (x^2 - 3)^3 + 27

Vedclass · Accepted Answer

$1$

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(C) The expression $2^{((x^2 - 3)^3 + 27)}$ attains its minimum value when the exponent $((x^2 - 3)^3 + 27)$ is minimized.Let $f(x) = (x^2 - 3)^3 + 27$.To find the minimum,we analyze the behavior of $(x^2 - 3)^3$.The term $(x^2 - 3)^3$ can take any real value because the range of $x^2 - 3$ is $[-3, \infty)$,and the cube function is strictly increasing.However,if we consider the domain of $x$ as all real numbers,$(x^2 - 3)^3$ can approach $-\infty$ as $x^2$ approaches $0$ (specifically,$(-3)^3 = -27$).Thus,the minimum value of $(x^2 - 3)^3 + 27$ is $(-27) + 27 = 0$.Therefore,the minimum value of the expression is $2^0 = 1$.

What is the minimum value of $2^{((x^2 - 3)^3 + 27)}$?

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