Video Example EXAMPLE 3 Find the local maximum and minimum values and saddle points of f(x, y) = x4 + y4 − 4xy + 1. SOLUTION We first locate the critical points:

(0, 0) is a saddle point

(1, 1) is a local minimum

(-1, -1) is another point  of local minimum

Step-by-step explanation:

We first locate the critical points. In order to get the critical points we need to find the first derivatives and then set them to zero.

f(x, y) = x⁴ + y⁴ - 4 xy + 1  

Find the first derivatives wrt to x and y

= 4x ³ - 4y  --> (1)

= 4y³ - 4x  --> (2)

Solve (1) for y

4x ³ - 4y = 0

x ³ = y

y = x ³ ---> (3)

Solve (2) for x

4y³ - 4x = 0

4y³ = 4x

y³ = x

x = y³  ----> (4)

Plug  (3) into (2)

4y³ - 4x

4(x ³)³ - 4x = 0

4x⁹ - 4x = 0

4x (x ⁸ - 1) = 0

4x (x ⁴ - 1)(x⁴  + 1) = 0

4x (x ² - 1)(x ² + 1)(x ⁴ + 1) = 0

4x (x - 1)(x + 1)(x ² + 1)(x ⁴ + 1) = 0

So

x = 1 , -1 , 0

In order to find values of y, Plug each x value in (3)

For

x = 0

corresponding y value

y = x ³

y = 0 ³

y = 0

For

x = 1

y = x ³

y = 1 ³

y = 1

For

x = -1

y = x ³

y = (-1 )³

y = -1

Hence we get the critical points which are:

(0, 0)

(1, 1)  and

(-1, -1)

Now for each critical point, we have to compute D(x,y)

For a critical point (x,y), D computed as:

D(x, y) = (x, y) - (x, y) - ( (x, y))²

After computing D(x,y) check:

If D(x, y) > 0 and (x, y) > 0:

f(x, y) is a local minimum.

If D(x, y) > 0 and   (x, y) < 0:

f(x, y) is a local maximum.

If D(x, y) < 0:

then f(x, y) is a saddle point

Here first compute the second derivative in order to get   ,   and

we have already computed:

= 4x ³ - 4y  --> (1)

= 4y³ - 4x  --> (2)

Now

 = 12x²

 = 12y²

 = -4

Compute D(x,y)

Critical Point (0, 0):

Putting values of   , and in above equation:

D(0,0) = 12(0)² * 12(0)² - (-4)²

          = 0 * 0 -16

D(0,0) = -16

We know that if D < 0, the critical point f(x, y) is a saddle point.

D(0,0) < 0 because D(0,0) = -16

Hence (0, 0) is a saddle point

Compute D(x,y)

Critical Point (1, 1):

Putting values of   , and in above equation:

D(1,1) = 12(1)² * 12(1)² - (-4)²

        = 12 * 12 - 16

D(1,1) = 128

We know that if D(x, y) > 0 and (x, y) > 0 then f(x, y) is a local minimum.

D(1,1) > 0 because D(1,1) = 128

(x, y) > 0 because   (x, y) = 12

Hence (1,1) is the local minimum

Compute D(x,y)

Critical Point (-1, -1):

Putting values of   , and in above equation:

D(-1,-1) = 12(-1)² * 12(-1)² - (-4)²

          = 12 * 12 - 16

D(-1,-1) = 128

We know that if D(x, y) > 0 and (x, y) > 0 then f(x, y) is a local minimum.

D(-1,-1) > 0 because D(-1,-1) = 128

(x, y) > 0 because   (x, y) = 12

Hence (-1,-1) is the local minimum

if x = 1, then the answer is 3.5

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.