Practicing Success
The least value of $\alpha \in R$ for which $4 \alpha x^2+\frac{1}{x} \geq 1$ for all $x>0$, is |
$\frac{1}{64}$ $\frac{1}{32}$ $\frac{1}{27}$ $\frac{1}{25}$ |
$\frac{1}{27}$ |
Let $f(x)=4 \alpha x^2+\frac{1}{x}, x>0$. Then, $f'(x)=8 \alpha x-\frac{1}{x^2}=\frac{8 \alpha x^3-1}{x^2}$ and, $f''(x)=8 \alpha+\frac{2}{x^3}$ At points of local maximum or minimum, we must have $f'(x)=0 \Rightarrow 8 \alpha x^3-1=0 \Rightarrow x=\left(\frac{1}{8 \alpha}\right)^{\frac{1}{3}}$ Clearly, $f''(x)=8 \alpha+\frac{2}{x^3}>0$ for all $x>0$ So, $f(x)$ attains its minimum value at $x=\left(\frac{1}{8 \alpha}\right)^{\frac{1}{3}}$ The minimum value of $f$ at $x=\left(\frac{1}{8 \alpha}\right)^{\frac{1}{3}}$ is $f\left(\left(\frac{1}{8 \alpha}\right)^{1 / 3}\right)=4 \alpha\left(\frac{1}{8 \alpha}\right)^{2 / 3}+2(8 \alpha)^{1 / 3}$ It is given that $f(x) \geq 1$ for all $x>0$. So, the minimum value of $f(x)$ is 1 ∴ $4 \alpha\left(\frac{1}{8 \alpha}\right)^{2 / 3}+(8 \alpha)^{1 / 3}=1 \Rightarrow 3 \alpha^{1 / 3}=1 \Rightarrow \alpha=\frac{1}{27}$ |