Please help me with this Let x denote a single observation o

Please help me with this:

Let x denote a single observation of a normal random variable, with mean=0 and unknown variance 2. Derive the maximum likelihood estimate for .

Solution

Given X~N(0,²), then L=(1/((2pie)))(exp^{((-x^2)/(2*2})))=>logL=((-1/2)*log(2pie))-((1/2)*log²)-((1/(2²))*(x²)), then the likelihood equation for estimate is dlogL/d=0,=>^=(x²).