We have an optimization problem with a linear constraint. Given that a plus 2b equals 1, we need to find the minimum value of the expression a to the fourth power over b plus 32 times b to the fourth power over a. Let's visualize the constraint line and identify the feasible region.
To solve this optimization problem, we use substitution. From the constraint a plus 2b equals 1, we can express a as 1 minus 2b. Substituting this into our objective function gives us a single-variable function f of b. The domain is restricted to 0 less than b less than one half, since both a and b must be positive.
To find the minimum, we take the derivative of our function and set it equal to zero. The derivative calculation involves the quotient rule and chain rule. After simplification, we find that the critical point occurs at b equals one sixth. This can be verified using symmetry arguments or by solving the derivative equation directly.
Now we calculate the minimum value by substituting b equals one sixth back into our expressions. When b equals one sixth, a equals two thirds. Substituting these values into the original objective function, we get sixteen eighty-firsts times six plus thirty-two over twelve ninety-six times three halves. After careful calculation, this simplifies to eleven ninths, which is our minimum value.
In conclusion, we have solved the constrained optimization problem. Given the constraint a plus 2b equals 1, the minimum value of a to the fourth power over b plus 32 times b to the fourth power over a is eleven ninths. This occurs when a equals two thirds and b equals one sixth. The result can be verified using alternative methods such as the AM-GM inequality or Lagrange multipliers.