If 4 x^2+8 x y+9 y^2-8 x-24 y+4=0, show that when (d y)/( d x)=0, x+y=1 and (d^2 y)/( d x^2)=(4)

step 1 Question number 7 4 x square plus 8 x5 plus 9 y squared minus 8 x minus 24 y plus 4 is equal to

If 4 x^2+8 x y+9 y^2-8 x-24 y+4=0, show that when (d y)/( d...

If $4 x^{2}+8 x y+9 y^{2}-8 x-24 y+4=0$, show that when $\frac{\mathrm{d} y}{\mathrm{~d} x}=0, x+y=1$ and $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{4}{8-5 y}$. Hence find the maximum and minimum values of $y$.

Key Concepts

Optimization on Implicit Functions

Optimization on implicitly defined functions involves using both the first and second derivative information without needing to solve explicitly for one variable. By leveraging conditions obtained from implicit differentiation and applying the second derivative test, one can determine the maximum and minimum values that the function attains, even when the relationship between the variables is not expressed in a standard function form.

Second Derivative Test

The second derivative test involves computing the second derivative of a function to determine the concavity at a critical point. If the second derivative is positive at a critical point, the function is locally convex, indicating a local minimum; if it is negative, the function is locally concave, indicating a local maximum. This test is essential when confirming the nature of extrema identified by the first derivative.

Implicit Differentiation

Implicit differentiation is a method used to find derivatives of functions defined by equations where the dependent and independent variables are intermingled. Instead of solving for one variable explicitly, both sides of the equation are differentiated with respect to the independent variable by applying the chain rule, allowing one to obtain dy/dx even when the function is given implicitly.

Critical Points Analysis

Critical points occur where the first derivative is equal to zero or is undefined. These points represent where the tangent to the curve is horizontal or vertical, and they are essential in identifying candidates for local maximum and minimum values in a function, particularly when the function is defined implicitly.

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