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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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