MCQ
If $a_1, a_2...,a_n$ an are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_1 + a_2 +.... + a_{n - 1} + 2a_n$ is
- ✓$n(2c)^{1/n}$
- B$(n + 1)c^{1/n}$
- C$2nc^{1/n}$
- D$(n + 1)(2c)^{1/n}$
$\Rightarrow \frac{a_{1}+a_{2}+\ldots+a_{n-1}+2 a_{n}}{n} \geq\left(a_{1} \cdot a_{2} \ldots a_{n-1} \cdot 2 a_{n}\right)^{1 / n}$
$\Rightarrow a_{1}+a_{2}+\ldots+2 a_{n} \geq n(2 c)^{1 / n}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$(A)$ $e-1$ $(B)$ $\int_1^e \ln (e+1-y) d y$ $(C)$ $e-\int_0^1 e^x d x$ $(D)$ $\int_1^r \ln y d y$