Question

The function $f(x) = x^2 \, \log \, x$ in the interval [1, e] has

• A. a point of maximum and minimum
• B. a point of maximum only
• C. no point of maximum and minimum in [1, e]
• D. no point of maximum and minimum

Answer 1

Answer: (C)

Given $f (x) = x^2 \, \log \, x$
Now for maximum and minimum f ' (x) = 0
$\Rightarrow \frac{x^{2}}{x} + 2x \log x = 0 $
$\Rightarrow x +2x \log x = 0 $
$\Rightarrow x\left(1+2 \log x\right) = 0 $
$\Rightarrow $ either $x = 0$ , or $ \log x = \frac{-1}{2}$ or $ x = e^{-1/2} $
But we have to search the minima and maxima in the interval [1, e]
$\therefore f"\left(x\right) = x . \frac{2}{x} + 1+2 \log x = 3 + 2 \log x $
Now at $x = 1$
$f " (1) = 3 = + ve$
$\Rightarrow \, f (x)$ is min. when $x = 1$
Therefore min. value $= 1^2 \, \log \, 1 = 0$
Now at $x = e$
$f "(e) = 3 + 2 = 5 (\because \, lne = 1)$ this is also + ve
But this is not possible that function has two min. values for two diff. values of x.
Thus, the function f(x) has no point of maximum and minimum in the interval [1, e].