Question 12

If x is a real number, then $$\sqrt{\log_{e}{\frac{4x - x^2}{3}}}$$ is a real number if and only if

Solution

$$\sqrt{\log_{e}{\frac{4x - x^2}{3}}}$$ will be real if $$\log_e\ \frac{\ 4x-x^2}{3}\ \ge\ 0$$

$$\frac{\ 4x-x^2}{3}\ >=\ 1$$

$$\ 4x-x^2-3\ >=\ 0$$

$$\ x^2-4x+3\ =<\ 0$$

1=< x=< 3

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