To find the greatest value of 16x²y³, we need to optimize the expression given the constraint 3x + 4y = 5. Let's go step by step to solve this problem:

Step 1: Rewrite the constraint equation:
To simplify the constraint equation, we can rewrite it as:
3x = 5 - 4y

Step 2: Substitute the rewritten constraint into the expression:
Now, substitute the rewritten constraint equation into the expression 16x²y³:
16(5 - 4y)²y³

Step 3: Expand and simplify:
Expand the expression and simplify it further:
16(25 - 40y + 16y²)y³
= 16(25y³ - 40y⁴ + 16y⁵)

Step 4: Differentiate the expression:
To find the maximum value, we need to differentiate the expression with respect to y:
d/dy (16(25y³ - 40y⁴ + 16y⁵))

Step 5: Apply the power rule of differentiation:
Differentiating each term using the power rule, we get:
16(75y² - 160y³ + 80y⁴)

Step 6: Set the derivative equal to zero:
To find the critical points, set the derivative equal to zero and solve for y:
75y² - 160y³ + 80y⁴ = 0

Step 7: Factor the equation:
Factor out y² from the equation:
y²(75 - 160y + 80y²) = 0

Step 8: Solve for y:
Set each factor equal to zero and solve for y:
y² = 0 or 75 - 160y + 80y² = 0

The first equation y² = 0 gives us y = 0.

Step 9: Solve the quadratic equation:
To solve the second equation, let's factor it further:
75 - 160y + 80y² = 0
5(15 - 32y + 16y²) = 0
5(4y - 1)(4y - 15) = 0

This gives us two possible values for y: y = 1/4 or y = 15/4.

Step 10: Find the corresponding x values:
Substitute the values of y back into the constraint equation to find the corresponding x values:
For y = 0: 3x + 4(0) = 5
3x = 5
x = 5/3

For y = 1/4: 3x + 4(1/4) = 5
3x + 1 = 5
3x = 4
x = 4/3

For y = 15/4: 3x + 4(15/4) = 5
3x + 15 = 5
3x = -10
x = -10/3

Step 11: Evaluate the expression for each point:
Now that we have three points (x, y), substitute

Answer is 3 can someone explain ???