For what real value of $b$ is the expression $\frac{1}{2}b^2 + 5b - 3$ minimized?
I notice that the expression is a quadratic function of $b$, and I recall that a quadratic function has a vertex that is either a minimum or a maximum point, depending on the sign of the leading coefficient. In this case, the leading coefficient is $\frac{1}{2}$, which is positive, so the vertex is a minimum point. To find the vertex, I can use the formula $b = -\frac{B}{2A}$, where $A$ and $B$ are the coefficients of the quadratic term and the linear term, respectively. In this case, $A = \frac{1}{2}$ and $B = 5$, so $b = -\frac{5}{2\cdot \frac{1}{2}} = -\frac{5}{1} = -5$. This means that the expression is minimized when $b = -5$. 