document.write( "Question 1158620: Locate the critical points of the following function. Then use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither.
\n" ); document.write( " f(x) = −e^x (x + 6) \n" ); document.write( "

Algebra.Com's Answer #781692 by solver91311(24713) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "I will presume that you have seen a proof that the product of two differentiable functions is, itself, differentiable, or if not, that you accept this fact on faith. Further, I assume that you accept the fact that

and

are both differentiable on all reals.\r
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\n" ); document.write( "\n" ); document.write( "Given the above, it is clear that there is no real number

such that

does not exist. Hence, the only critical points are where

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\n" ); document.write( "\n" ); document.write( "And by inspection, we see that only

, hence

is the only critical point of the function.\r
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\n" ); document.write( "\n" ); document.write( "I leave it as an exercise for you to show that the second derivative is

, determine the sign of the value of

, and use that result to select local maxima or minima based on:\r
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Local maxima\r
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\ 0\ \Right\ \"> Local minima\r
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\n" ); document.write( "John
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\n" ); document.write( "My calculator said it, I believe it, that settles it
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