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What is the smallest value of a for which the inequality $$5\ln(x)-5x^2≤4x+a$$ is observed for al $$x>0$$
My progress:
I rewrite it as $$5\ln(x)-5x^2-4x≤a$$ and then derivate andfind the ciritical point
$$f'(x)=\frac{5}{x}-10x-4$$
$$x_1=\frac{1}{10}(-2-3\sqrt{6})$$ (Notice that it says $$x>0$$ and this is negative root so we shall ignore it.
$$x_2=\frac{1}{10}(3\sqrt{6}-2)$$ (this root work fine!)
then I shall put that x value in $$5\ln(x)-5x^2-4x$$ and I get the answer
http://www.wolframalpha.com/input/?i=5*ln%281%2F10%283sqrt%286%29-2%29%29-5%281%2F10%283sqrt%286%29-2%29%29^2-4%281%2F10%283sqrt%286%29-2%29%29
Is this correct thinking or I am doing wrong?
My progress:
I rewrite it as $$5\ln(x)-5x^2-4x≤a$$ and then derivate andfind the ciritical point
$$f'(x)=\frac{5}{x}-10x-4$$
$$x_1=\frac{1}{10}(-2-3\sqrt{6})$$ (Notice that it says $$x>0$$ and this is negative root so we shall ignore it.
$$x_2=\frac{1}{10}(3\sqrt{6}-2)$$ (this root work fine!)
then I shall put that x value in $$5\ln(x)-5x^2-4x$$ and I get the answer
http://www.wolframalpha.com/input/?i=5*ln%281%2F10%283sqrt%286%29-2%29%29-5%281%2F10%283sqrt%286%29-2%29%29^2-4%281%2F10%283sqrt%286%29-2%29%29
Is this correct thinking or I am doing wrong?