Question

Find the intervals of monotonicity of the function : $y=\frac{10}{4x^3-9x^2+6x}$

• A. decrease in $(-\infty ,0)\cup (0,\frac{1}{2})\cup (2,\infty )$ increases in $\left(\frac{1}{2},2\right)$
• B. decrease in $(-\infty ,0)\cup (0,\frac{1}{2})\cup (1,\infty )$ increases in $\left(\frac{1}{2},1\right)$
• C. decrease in $(-\infty ,0)\cup (0,\frac{1}{3})\cup (1,\infty )$ increases in $\left(\frac{1}{3},1\right)$
• D. decrease in $(-\infty ,0)\cup (0,\frac{2}{3})\cup (1,\infty )$ increases in $\left(\frac{2}{3},1\right)$

Answer 1

The correct option is B decrease in $(-\infty ,0)\cup (0,\frac{1}{2})\cup (1,\infty )$ increases in $\left(\frac{1}{2},1\right)$

For monotonically increasing function,
$y^{\prime }>0$
Hence, $-10(12x^2-18x+6)>0$
$2x^2-3x+1<0$
$2x^2-2x-x+1<0$
$2x(x-1)-(x-1)<0$
$(2x-1)(x-1)<0$
$xϵ(\frac{1}{2},1)$
For monotonically decreasing
$xϵ(-\infty ,\frac{1}{2})\cup (1,\infty )-\left\{0\right\}$.