91Ó°ÊÓ

Partial derivatives are used to find the rate of change of a multivariable function with respect to one variable while keeping other variables constant. In our function, we have two variables, "t" and "p". To find the critical points of the function \( G(t, p) = p e^t - 3p \), we need to calculate the first partial derivatives with respect to both variables.
To compute the partial derivative of \( t \), labeled as \( G_t(t, p) \), we differentiate the function \( G(t, p) \) while considering \( p \) as a constant. The result is \( G_t(t, p) = p e^t \).
Similarly, to compute the partial derivative with respect to \( p \), denoted as \( G_p(t, p) \), we take the derivative with \( t \) held constant, resulting in \( G_p(t, p) = e^t - 3 \).

• Partial derivatives help determine how the function output changes as one of the variables is changed.
• Finding where these derivatives equal zero helps in locating the critical points of the function.

Once we have the critical points, to classify them, we utilize the Hessian matrix, which involves second-order partial derivatives. The Hessian determinant is a specific value derived from this matrix, calculated as \( H = G_{tt}G_{pp} - (G_{tp})^2 \).
For our function, the second partial derivatives needed are:
\[ G_{tt}(t, p) = p e^t, \quad G_{tp}(t, p) = e^t, \quad G_{pp}(t, p) = 0 \]
Evaluating these at the critical point \( (t, p) = (\ln(3), 0) \), we find:
\[ G_{tt}(\ln(3), 0) = 0, \quad G_{tp}(\ln(3), 0) = 3, \quad G_{pp}(\ln(3), 0) = 0 \]
Substituting in the Hessian determinant formula yields \( H = 0 \cdot 0 - 3^2 = -9 \).

• The determinant gives insight into the nature of the critical point: positive implies local minima or maxima, zero is inconclusive, while negative indicates a saddle point.
• In this case, a negative Hessian confirms the critical point is a saddle point.

A saddle point is a type of critical point where the function does not have a local minimum or maximum. Instead, it acts as a sort of "pass" between hills or valleys, where the function changes direction from increasing to decreasing or vice versa.
In the context of the function \( G(t, p) = p e^t - 3p \), the Hessian determinant calculated was \( -9 \) at the critical point \( (\ln(3), 0) \). This negative value directly points to the presence of a saddle point.

• Saddle points occur when the surface defined by the function is curved upwards in one direction and downwards in a perpendicular direction.
• Graphically, you might picture a saddle point like the dip in a saddle, hence the term.
• Identifying saddle points is important for understanding the behavior of functions in optimization problems.