The nth derivative of x e^x vanishes when - Vidyadip

MCQ

The nth derivative of $x{e^x}$ vanishes when

A
$x = 0$
B
$x = - 1$
✓
$x = - n$
D
$x = n$

✓

Answer

Correct option: C.

$x = - n$

c
(c) $f(x) = x{e^x}$
$f'(x) = {e^x} + x{e^x}$
$f''(x) = {e^x} + {e^x} + x{e^x} = 2{e^x} + x{e^x}$
$f'''(x) = 2{e^x} + {e^x} + x{e^x} = 3{e^x} + x{e^x}$
……………………………………………
……………………………………………
${f^n}(x) = n{e^x} + x{e^x}$.

Now, ${f^n}(x) = 0$

==> $n{e^x} + x{e^x} = 0$ ==>$x = - n$.

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