If `M` be the maximum value of `72 int _(0) ^(y) sqrt(x ^(4) +(y-y^(2))^(2))dx` for `y in [0,1],` then find `M/6`.

Answer

To solve the problem, we need to find the maximum value \( M \) of the function defined by the integral:

\[ M = 72 \int_{0}^{y} \sqrt{x^4 + (y - y^2)^2} \, dx \]

for \( y \) in the interval \([0, 1]\), and then find \( \frac{M}{6} \).

### Step 1: Define the function
Let:
\[ f(y) = 72 \int_{0}^{y} \sqrt{x^4 + (y - y^2)^2} \, dx \]

### Step 2: Differentiate \( f(y) \)
To find the maximum value, we first differentiate \( f(y) \) with respect to \( y \):
\[ f'(y) = 72 \cdot \sqrt{y^4 + (y - y^2)^2} \]
(using the Fundamental Theorem of Calculus).

### Step 3: Set the derivative to zero
To find critical points, we set \( f'(y) = 0 \):
\[ \sqrt{y^4 + (y - y^2)^2} = 0 \]
This implies:
\[ y^4 + (y - y^2)^2 = 0 \]

### Step 4: Simplify the equation
Expanding \( (y - y^2)^2 \):
\[ (y - y^2)^2 = y^2 - 2y^3 + y^4 \]
Thus, we have:
\[ y^4 + y^2 - 2y^3 + y^4 = 0 \]
which simplifies to:
\[ 2y^4 - 2y^3 + y^2 = 0 \]
Factoring out \( y^2 \):
\[ y^2(2y^2 - 2y + 1) = 0 \]

### Step 5: Analyze the factors
The factor \( y^2 = 0 \) gives \( y = 0 \). The quadratic \( 2y^2 - 2y + 1 = 0 \) has a discriminant:
\[ (-2)^2 - 4 \cdot 2 \cdot 1 = 4 - 8 = -4 \]
Since the discriminant is negative, \( 2y^2 - 2y + 1 \) has no real roots. Thus, \( f'(y) \) is always non-negative for \( y \in [0, 1] \), indicating that \( f(y) \) is an increasing function.

### Step 6: Evaluate \( f(y) \) at the endpoints
Since \( f(y) \) is increasing, the maximum value occurs at the upper limit \( y = 1 \):
\[ M = f(1) = 72 \int_{0}^{1} \sqrt{x^4 + (1 - 1^2)^2} \, dx \]
This simplifies to:
\[ M = 72 \int_{0}^{1} \sqrt{x^4 + 0} \, dx = 72 \int_{0}^{1} x^2 \, dx \]

### Step 7: Calculate the integral
The integral \( \int_{0}^{1} x^2 \, dx \) is:
\[ \int x^2 \, dx = \frac{x^3}{3} \Big|_{0}^{1} = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \]
Thus:
\[ M = 72 \cdot \frac{1}{3} = 24 \]

### Step 8: Find \( \frac{M}{6} \)
Finally, we compute:
\[ \frac{M}{6} = \frac{24}{6} = 4 \]

### Final Answer
\[ \frac{M}{6} = 4 \]