To determine if the given point is a maximum of the function F, we need to calculate the directional derivative of F at this point and perform a second derivative test.
The directional derivative of the function F at the point (4,2) can be calculated by deriving F relative to R and S. According to the computed function:
The derivative of F with respect to R at the point (4,2) is -0.392.
The derivative of F with respect to S at the point (4,2) is -1.106.
Thus, the gradient vector of F at the point (4,2) is (-0.392, -1.106), which points in the direction of greatest function's increase.
To perform the second derivative test, we need to calculate the second order partial derivatives. The second order derivatives according to the computed function are:
The second derivative of F with respect to R at the point (4,2) is 0.034.
The second derivative of F with respect to S at the point (4,2) is -0.068.
The derivative of F_R with respect to S (F_RS) at the point (4,2) is -0.268.
Also, the derivative of F_S with respect to R (F_SR) at the point (4,2) is -0.268.
In the second derivative test, we compute the determinant of the Hessian matrix - the matrix of the second order partial derivatives - at the point of interest. The determinant of the Hessian matrix at the point (4,2) is -0.074136.
By applying the second derivative test which states that if the determinant of the Hessian matrix is greater than 0 and the second derivative with respect to R and S are both less than 0, the point is a local maximum, we find that, in this case, the function F does not have a local maximum at the point (4,2), since the determinant is negative.
So, combining the results, we conclude that the function F=4.2+0.008R+0.102S+0.017R²-0.034S²-0.268RS does not have a local maximum at the point (4,2).