Answer:
a) [tex]P (x,y) \approx (0.1,1.05)[/tex], b) [tex]P(x,y) = (0.07,1.04)[/tex].
Step-by-step explanation:
a) The graphic is enclosed to the problem. By visual inspection, an absolute maximum is found.
[tex]P (x,y) \approx (0.1,1.05)[/tex]
b) The exact method consists in the application of the First and Second Derivative Tests. First and second derivatives are, respectively:
[tex]f'(x) = 1 - 14\cdot x[/tex]
[tex]f''(x) = -14[/tex]
The First Derivative Test consists in equalizing the first derivative to zero and solving the expression:
[tex]1 - 14\cdot x = 0[/tex]
[tex]x = 0.07[/tex]
According to the second derivative, the critical point leads to a maximum. The remaining component is determined by evaluation the polynomial:
[tex]y = 1 +0.07-7\cdot (0.07)^{2}[/tex]
[tex]y = 1.04[/tex]
The exact solution is [tex]P(x,y) = (0.07,1.04)[/tex], indicating that graphic solution leads to a good approximation.
