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Andrew [12]
3 years ago
9

Local versus absolute extrema. If you recall from single-variable calculus (calculus I), if a function has only one critical poi

nt, and that critical point is a local maximum (or say local minimum), then that critical point is the global/absolute maximum (or say global/absolute minnimum). This fails spectacularly in higher dimensions (and thereís a famous example of a mistake in a mathematical physics paper because this fact was not properly appreciated.) You will compute a simple example in this problem. Let f(x; y) = e 3x + y 3 3yex . (a) Find all critical points for this function; in so doing you will see there is only one. (b) Verify this critical point is a local minimum. (c) Show this is not the absolute minimum by Önding values of f(x; y) that are lower than the value at this critical point. We suggest looking at values f(0; y) for suitably chosen y
Mathematics
1 answer:
stich3 [128]3 years ago
7 0

Answer:

Step-by-step explanation:

Given that:

a)

f(x,y) = e^{3x} + y^3 - 3ye^x \\ \\ \implies \dfrac{\partial f}{\partial x} = 0 = 3e^{3x} -3y e^x = 0 \\ \\  e^{2x}= y \\ \\ \\  \implies \dfrac{\partial f}{\partial y } = 0 = 3y^2 -3e^x  = 0  \\ \\ y^2 = e^x

\text{Now; to determine the critical point:}- f_x = 0 ;  \ \ \ \ \  f_y =0

\implies  e^{2x} = y^4  = y  \\ \\ \implies y = 0 \& y =1  \\ \\ since y \ne 0 , \ \ y = 1,  \ \ x= 0\\\text{Hence, the only possible critical point= }(0,1)

b)

\delta = f_xx, s = f_{xy}, t = f_{yy} \\ \\ . \  \ \ \ \ \ \ \  D = rt-s^2 \\ \\  i)  Suppose D >0 ,\ \ \ r> 0 \ \text{then f is minima} \\ \\  ii)  Suppose  \ D >0 ,\ \ \ r< 0 \ \text{then f is mixima} \\ \\ iii) \text{Suppose  D} < 0 \text{, then  f is a saddle point} \\ \\  iv) Suppose  \ D = 0 \ \  No \ conclusion

Thus  \ at (0,1) \\ \\ \delta = f_{xx}  = ge^{3x}\implies \delta (0,1) = 6 \\ \\ S = f_{xy} = -3e^x  \\ \\ \implies S_{(0,1)} = -3 \\ \\ t = f_{yy} = 6y  \\ \\

\implies t_{0,1} = 6

Now; D = rt - s^2 \\ \\  = (6)(6) -(-3)^2

= 36 - 9 \\ \\ = 27 > 0 \\ \\  r>0

\text{Hence, the critical point} \ (0,1)  \ \text{appears to be the local minima}

c)

\text{Suppose we chose x = 0 and y = -3.4} \\ \\ \text{Then, we have:} \\ \\ f(0,-3.4) = 1+ (-3.4)^3 + 3(3.4) \\ \\ = -28.104 < -1

\text{However, if  f (0,1) = 1 +1 -3 = -1 \\ \\ f(0,-3.4) = -28.104} < -1}  \\ \\  \text{This explains that} -1 \text{is not an absolute minimum value of f(x,y)}

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ipn [44]
There are 2 way to solve this.

one using Pythagoras theorem and 2nd using trigonometry

so lets solve it by both

using Pythagoras theorem we know

base^2 + perpendicular^2 = hypotanes^2

6^2 + x^2 = 12^2

36 + x^2 = 144

x^2 = 144- 36 = 108

x = √(108) = √( 2×2×3×3×3)

= (2×3) √ (3) = 6 √3

so answer is option 2

bow lets use trigonometry

we know
sin theta = perpendicular / hypotanes
sin 60 = x /12
x = 12 × sin 60
we kNow sin 60 = √3/ 2
so
x = 12×√3 /2 = 6√3
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3 years ago
Can anyone teach plsssss........................ Given that PR= 17cm and SR= 6cm.Find the length, in cm of TS. ​
Lemur [1.5K]

Answer:

9cm

Step-by-step explanation:

  1. Since PQST is a rectangle, ∠QST=∠QSR=∠90º. So triangle QSR is a right triangle.
  2. By the Pythagorean Theorem and that SR=6 and QR=10, we can say that 6^2+b^2=10^2. So, 36+b^2=100. And b^2=64 and b=8. So QS=8.
  3. Now we know that QS=PT=8.
  4. By the Pythagorean Theorem and that PR=17 and PT=8, we can say a^2+8^2=17^2. So, a^2+64=289. And a^2=225 and a=15.
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Kim and jay leave at the same time to travel 25 miles to the beach. Kim drives 9 miles in 12 minutes. Jay drives 10 miles in 15
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Find the line through (3, 1, −2) that intersects and is perpendicular to the line x = −1 + t, y = −2 + t, z = −1 + t. (HINT: If
mel-nik [20]

Answer:

( xo , yo , zo ) = ( 1 , 0 , 1 )

Step-by-step explanation:

Given:-

- A line passing through point (3, 1, −2) intersects and is perpendicular to line with coordinates:

                  x = −1 + t, y = −2 + t, z = −1 + t   .... t = arbitrary parameter.

Find:-

The coordinates for point of intersection.

Solution:-

- The line that passes through point (3, 1, −2) = ( a, b , c ) and an a arbitrary point on the given line have the following direction vector d2 :

                 d2 = ( x2 , y2 , z2 )

                 x2 = a - ( x ) = 3 - ( -1 + t ) = 4 - t

                 y2 = b - ( y ) = 1 - ( -2 + t ) = 3 - t

                 z2 = c - ( z )  = -2 - ( -1 + t ) = -1 - t

                d2 = (  4 - t , 3 - t , -1 - t )

- The direction vector d1 of the given line is:

                 d1 = ( x1 , y1 , z1 )

                 x1 = 1

                 y1 = 1

                 z1 = 1

                 d1 = ( 1 , 1 , 1 )

- The dot product of two orthogonal vectors is always equal to zero:

                 d1.d2 = 0

                 (  4 - t , 3 - t , -1 - t ) . ( 1 , 1 , 1 ) = 0

- Solve for parameter (t):

                 (4 - t) + (3 - t) + (-1 - t) = 0

                  6 -3t = 0

                  t = 2  

- The coordinates of the point of intersections can be evaluated by substituting the value of "t" into the given equation of line:

                 xo ( t = 2) = - 1 + 2 = 1

                 yo ( t = 2) = - 2 + 2 = 0

                 zo ( t = 2) = - 1 + 2 = 1

- The coordinates are:

                 ( xo , yo , zo ) = ( 1 , 0 , 1 )

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