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Answer by Eric Towers for Finding the minimum value of a function without...
First, the minimum value of $x$ is $b$ such that $f(x) - b$ has a (double) root. (That is, the amount you must shift the graph of $f$ down so that it meets the $x$-axis once.) So we are looking for a...
View ArticleAnswer by farruhota for Finding the minimum value of a function without using...
Use AM-GM:$$x^4+\frac1{x^2}=x^4+\frac1{2x^2}+\frac1{2x^2}\ge 3\sqrt[3]{\frac1{4}},$$equality occurs when $x^4=\frac1{2x^2}=\frac1{2x^2} \Rightarrow x=\pm\frac1{\sqrt[6]{2}}$.
View ArticleAnswer by lab bhattacharjee for Finding the minimum value of a function...
Hint:As $x^4,1/x^2>0$ using Weighted Form of Arithmetic Mean-Geometric Mean Inequality $$\dfrac{ax^4+bx^{-2}}{a+b}\ge\sqrt[a+b]{x^{4a-2b}}$$Set $4a-2b=0\iff b=2a$
View ArticleFinding the minimum value of a function without using Calculus
Find the minimum value the function $f(x) = x^4 + \frac{1}{x^2}$ when $x \in \Bbb R^*$My attempt:Finding the minimum value of this function using calculus is a piece of cake. But since this question...
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