<s>[INST] Calculate $f(x)$ at a specific point Question:
Calculate $f(x) = \frac{49}{x^2} + x^2$ at points for which $\frac{7}{x}+x =3$
My attempt:-
I tried to find the value of $x$ and insert in $f(x)$
$$\frac{7}{x}+x =3$$
$$7+x^2 =3x$$
$$x^2 -3x + 7=0$$
$$x = \frac{3+\sqrt{9-7*4}}{2}$$
Now $x$ is coming out to be irrational and things get a lot more difficult from there. What should I do?
 [/INST] For $x \not =0$, $x$ real:
1) No real, negative $x$ satisfies above equation.
2) For $x >0$ :AM-GM:
$(7/x) +x \ge 2\sqrt{7} \gt 4 $.
Hence there is no real $x$ with: $(7/x) +x = 3$ .
Left to do, find a complex solution.
Note:
$(7/x+x)^2 =$
$ 49/x^2 +2(7) + x^2 =9.$
$49/x^2 +x^2 =-5.$
Hence $f(x)=-5.$
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