Question

Maximum value of x3y3 + 3 x*y when x+y = 8.

Answer 1

Answer:

The answer is 4144.

Step-by-step explanation:

We have to find the maximum value of f(x)= $x^{3}y^{3}+3xy$ when $x+y=8$

We can write $x+y=8$ as $y=8-x$

Substituting the value of y

$f(x)=x^{3}(8-x)^{3} +3x(8-x)$

= $3x^{2}(8-x)^{2}[-x+8-x]+3[-x+8-x]$

= $3(8-2x)[x^{2}(8-x)^{2}+1]$

For maximum, we will equate the equation to 0.

And we get,

$8-2x=0$ => x = 4

And as $y=8-x$

$y =8-4= 4$ > y = 4

So, we will put these values in equation to get maximum value.

= $x^{3}y^{3}+3xy$

= $xy(x^{2}y^{2}+3)$

= $4\times4((16\times16) +3)$

= $16(259)$ = 4144

Answer 2

Maximum value of x³y³ + 3 x.y when x + y = 8 is 4144

Further explanation

To solve this problem there are several basic principles in Derivatives that need to be recalled, namely:

$y = a ~ x^n \Rightarrow \frac{dy}{dx} = a ~ n ~ x^{n-1}$

$y = \sin x \Rightarrow \frac{dy}{dx} = \cos x$

$y = \cos x \Rightarrow \frac{dy}{dx} = - \sin x$

$y = u \times v \Rightarrow \frac{dy}{dx} = u' \times v + u \times v'$

$y = u \div v \Rightarrow \frac{dy}{dx} = \frac{u' \times v - u \times v'}{v^2}$

$y = u^n \Rightarrow \frac{dy}{dx} = n \times u^{n-1} \times u'$

$\text{where u and v are functions in variable x}$

$\text{and u' and v' are derivatives of u and v}$

Let us now tackle the problem !

Let :

$z = x^3y^3 + 3xy$

If x + y = 8 → y = 8 - x , then

$z = x^3(8-x)^3 + 3x(8-x)$

$\frac{dz}{dx} = 3x^2(8-x)^3 - x^33(8-x)^2 + 3(8-x) + 3x(-1)$

$0 = x^2(8-x)^3(3(8-x) -3x) + (3(8-x) - 3x)$

$0 = (x^2(8-x)^3 + 1)(3(8-x) -3x))$

$3(8-x) - 3x = 0$

$3(8-x) = 3x$

$8 - x = x$

$8 = x + x$

$8 = 2x$

$x = \frac{8}{2}$

$\boxed {x = 4}$

$x + y = 8$

$4 + y =8$

$y = 8 - 4$

$\boxed {y = 4}$

$z = x^3y^3 + 3xy$

$z = 4^34^3 + 3(4)(4)$

$z = 4^34^3 + 3(4)(4)$

$\large {\boxed {z = 4144} }$

Learn more

• Implicit Differentiation : brainly.com/question/4711711
• Logarithmic Differentiation : brainly.com/question/9226310
• Calculus Problem : brainly.com/question/11237537

Answer details

Grade: High School

Subject: Mathematics

Chapter: Differentiation

Keywords: Maximum , Minimum , Value , Function , Variable , Derivation , Differentiation