Lesson Finding the minimum of a function defined on a curve in a coordinate plane

This Lesson (Finding the minimum of a function defined on a curve in a coordinate plane) was created by by ikleyn(52944)

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Finding the minimum of a function defined on a curve in a coordinate plane

Problem 1

The positive variables x and y are such that x%5E4y

= 32.
A function z is defined by z = x%5E2+%2B+y

.
Find the values of x and y that give z a stationary value and show that this value of z is a minimum.

Solution

 --> 

 =        (1)

 =             (2)

 =             (3)


The stationary point is where the derivative is zero.

 = 

 = 

 = 

x = 2    (actually, x = +/- 2, but since we consider everything in positive numbers, we take x = 2).

At the stationary point,  and  =  =  = 


    The stationary point is a minimum if the second derivative at the point is positive; 
    or it is a maximum if that derivative is negative.  
    At x = 2, the second derivative is OBVIOULSLY positive (it is clear without any calculations)


So the stationary point is a minimum.


ANSWERS: z has a stationary point that is a minimum when x = 2 and y = 2.


To make this result more visible and visually verifiable, I prepared a plot below.



    

            Plot  z =  +   (see formula (1)