Find min. value of y = 8x exp^(x) on interval [-2, 0]

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ladyazpy

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Find the minimum value of y=8xexp^(x) on the interval [-2,0]

first i applied the product rule so

= 8x[exp^(x)] + [exp^(x)] (8)

= exp^(x){8x+8}

now on this part i have to equal them to zero to find my critical points however it confusing

Critical points

exp^(x)=0 ; 8x+8=0

Critical points : 0, -1

Now i have to substitute my ciritical points and given interval

f(-2) 8(-2)exp^(-2)= -2.17
f(-0) 8(0)exp^(0)= 0
f(-1) 8(-1)exp^(-1)=-2.94

So my minimum value is -2.94 when x=-1
is this right????? please check my work
 
Re: Exponential function!

Hello, ladyazpy!

Your final answer is correct, but there is a small error . . .

Find the minimum value of \(\displaystyle y\:=\:8xe^x\) on the interval \(\displaystyle [-2,0]\)

First i applied the product rule so: \(\displaystyle \:y' \:=\:8xe^x\,+\,e^x\cdot8 \:=\:e^x(8x\,+\,8)\)

now i equate it to zero to find my critical points: \(\displaystyle \:e^x(8x\,+\,8)\:=\:0\)

Critical points: \(\displaystyle \:0,\:-1\) . . . . no
Click to expand...

We have: \(\displaystyle \:8x\,+\,8\:=\:0\;\;\Rightarrow\;\;x\,=\,-1\)

. . but \(\displaystyle e^x\,=\,0\) has no solution.

So \(\displaystyle x\,=\,0\) is not a critical point.


But you are correct to check the three values:
. . the critical value and the endpoints of the interval.

. . \(\displaystyle \begin{array}{ccccccc}f(-2) & = & -16e^{-2} & = & -2.17 & & \\ f(-1) & = & -8e^{-1} & = & -2.94 & \;\Leftarrow\; & \text{minimum}\\ f(0) & = & 0e^0 & = & 0 & &\end{array}\)

 
Re: Exponential function!

ladyazpy said:
Find the minimum value of y=8xexp^(x) on the interval [-2,0]

first i applied the product rule so

= 8x[exp^(x)] + [exp^(x)] (8)

= exp^(x){8x+8}

now on this part i have to equal them to zero to find my critical points however it confusing

Critical points

exp^(x)=0 ; 8x+8=0

Critical points : 0, -1 ... why is x = 0 a critical point

Now i have to substitute my ciritical points and given interval

f(-2) 8(-2)exp^(-2)= -2.17
f(-0) 8(0)exp^(0)= 0
f(-1) 8(-1)exp^(-1)=-2.94

So my minimum value is -2.94 when x=-1
is this right????? please check my work
Click to expand...
 
When requested to show your work, please reply within that thread, rather than starting a new one.

Thank you.

Eliz.
 
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