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Mariana [72]
1 year ago
5

Find the minimum value if f(x) = xe^x over [-2,0]

Mathematics
1 answer:
SashulF [63]1 year ago
6 0

Given function:

f(x)=xe^x

The minimum value of the function can be found by setting the first derivative of the function to zero.

f^{\prime}(x)=xe^x+e^x\begin{gathered} xe^x+e^x\text{ = 0} \\ e^x(x\text{ + 1)  = 0} \end{gathered}

Solving for x:

\begin{gathered} x\text{ + 1 = 0} \\ x\text{ = -1} \end{gathered}\begin{gathered} e^x\text{ = 0} \\ \text{Does not exist} \end{gathered}

Substituting the value of x into the original function:

\begin{gathered} f(x=1)=-1\times e^{^{-1}}_{} \\ =\text{ -}0.368 \end{gathered}

Hence, the minimum value in the given range is (-1, -0.368)

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==================================================

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This logic is used to determine the other three corner points as well. The upper right corner is point D since AD and DC have D in common. The same goes for the bottom points as well.

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