# LMVT and Mean value theorem for integrals.

I am confused whether to use Lagrange's Mean value theorem (LMVT) or normal Mean value for integrals to find the mean value.

For example :- $f(x) = x^2 + 3x + 2$ in the interval $[1,4].$

I get the value of the mean as 2.5 using LMVT and value of mean or average value of function as 2.593 using mean value for integrals method.

Why am I getting different values ?