find the absolute minimum and absolute maximum of\n\n$f(x,y)=18 - 5x + 8y$\n\non the closed triangular…

find the absolute minimum and absolute maximum of\n\n$f(x,y)=18 - 5x + 8y$\n\non the closed triangular region with vertices $(0,0),(8,0)$ and $(8,13)$.\n\nlist the minimum/maximum values as well as the point(s) at which they occur. if a min or max occurs at multiple points separate the points with commas.\n\nminimum value:\n\noccurs at\n\nmaximum value:\n\noccurs at\n\nsubmit answer next item

find the absolute minimum and absolute maximum of\n\n$f(x,y)=18 - 5x + 8y$\n\non the closed triangular region with vertices $(0,0),(8,0)$ and $(8,13)$.\n\nlist the minimum/maximum values as well as the point(s) at which they occur. if a min or max occurs at multiple points separate the points with commas.\n\nminimum value:\n\noccurs at\n\nmaximum value:\n\noccurs at\n\nsubmit answer next item

Answer

Explanation:

Step1: Analyze the function on the boundaries

  • Boundary 1: (y = 0), (0\leq x\leq8) The function becomes (f(x,0)=18 - 5x). Since the derivative (f^\prime(x,0)=-5<0), it is a decreasing function. At (x = 8), (f(8,0)=18-5\times8=18 - 40=-22); at (x = 0), (f(0,0)=18).
  • Boundary 2: (x = 8), (0\leq y\leq13) The function becomes (f(8,y)=18-5\times8 + 8y=8y - 22). Since the derivative (f^\prime(8,y)=8>0), it is an increasing function. At (y = 0), (f(8,0)=-22); at (y = 13), (f(8,13)=8\times13-22=104 - 22 = 82).
  • Boundary 3: Find the equation of the line passing through ((0,0)) and ((8,13)) The slope (m=\frac{13-0}{8 - 0}=\frac{13}{8}), and the equation is (y=\frac{13}{8}x). Substitute (y=\frac{13}{8}x) into (f(x,y)): (f(x,\frac{13}{8}x)=18-5x + 8\times\frac{13}{8}x=18-5x+13x=18 + 8x). Since the derivative (f^\prime(x,\frac{13}{8}x)=8>0), it is an increasing function. At (x = 0), (f(0,0)=18); at (x = 8), (f(8,13)=82).

Step2: Check the critical points

Find the partial derivatives: (\frac{\partial f}{\partial x}=-5), (\frac{\partial f}{\partial y}=8). Since (\frac{\partial f}{\partial x}\neq0) and (\frac{\partial f}{\partial y}\neq0) simultaneously, there are no critical points in the interior of the triangular region.

Answer:

Minimum value: (-22) Occurs at ((8,0)) Maximum value: (82) Occurs at ((8,13))