I need some help with the following problem. Find the maximum value of the function $f(x)=|3x^2+2ax-1|$ for $x\in[-1,1]$ if $-2 \leq a \leq 2$.
Maximum value of a function
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analysis
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0Honestly, I h$a$ve not seen or noted such this claim (or fact) before. – 2012-11-28
1 Answers
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Consider the function $f(x,a)=3x^2+2ax-1$. The maximum of this function is the same as for $|f|$. We can consider also a region: $-2 \leq a \leq 2, \;\; -1 \leq x \leq 1$ By the method which you do know that we find the origin as a interior critical point in which the function $f$ vanishes. Now, we consider the border which are $x=-1,x=1,a=-2,a=2$. A simple calculating shows us that at the point $(x,a)=(1,2)$ or $(-1,-2)$ the function reaches at $6$. and other points along side these borders are $(2/3,-2),(-2/3,2)$ which make the value $-7/3$. So the max would be $6$.
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0Why a and x should be positive or negative at the same time? – 2012-11-28