exponential equation(2)

this is the easiest of ur three exponential equations..

let x/3 = t

the equation becomes

2012^3t = 2011^3t + 3*2011^2t + 3* 2011^t

≡(2012)^3t = (2011^t+ 1)³

iff (2012)^t= (2011)^t + 1

By the monotonicity of the function 2012^t- 2011^t we conclude t=1 is the unique
solution.

so x= 3t = 3

$Shuphodip Right Answer..\\\\ 2012^t=2011^t+1\Leftrightarrow 2011^t+1=2012^t$\\\\ \left(\frac{2011}{2012}\right)^t+\left(\frac{1}{2012}\right)^t=1\Leftrightarrow \left(\frac{2011}{2012}\right)^t+\left(\frac{1}{2012}\right)^t-1$\\\\\\ Now Let $f(t)=\left(\frac{2011}{2012}\right)^t+\left(\frac{1}{2012}\right)^t-1$\\\\\\ So function $f(x)$ is Strictly Decreasing function bcz $f^{'}(x)<0\forall x\in \mathbb{R}$.\\\\ So It Cut X-axis at exactly one Point.\\\\ Means Unique solution of the Given equation.

$bcz $\frac{d}{dx}(f(x))<0\forall x\in \mathbb{R}$.\\\\ So It Cut X-axis at exactly one Point.\\\\ Means Unique solution of the Given equation.