Question

For what values of xx is the graph of y = xe^−2x concave down?

values =
(Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10) .)

Answer 1

raph of function f(x) is concave down when f"(x) < 0

�

=

�

�

−

2

�

y=xe

−2x

Differentiating with respect to x

�

′

=

�

−

2

�

+

�

⋅

�

−

2

�

(

−

2

)

y

′

=e

−2x

+x⋅e

−2x

(−2)

⇒

�

′

=

�

−

2

�

(

1

−

2

�

)

⇒y

′

=e

−2x

(1−2x)

Differentiating with respect to x again

�

′

′

=

�

−

2

�

(

0

−

2

)

+

(

1

−

2

�

)

�

−

2

�

⋅

(

−

2

)

y

′′

=e

−2x

(0−2)+(1−2x)e

−2x

⋅(−2)

⇒

�

′

′

=

�

−

2

�

(

−

2

−

2

(

1

−

2

�

)

)

⇒y

′′

=e

−2x

(−2−2(1−2x))

⇒

�

′

′

=

�

−

2

�

(

−

2

−

2

+

4

�

)

⇒y

′′

=e

−2x

(−2−2+4x)

⇒

�

′

′

=

(

4

�

−

4

)

�

−

2

�

⇒y

′′

=(4x−4)e

−2x

point of inflection when

�

′

′

=

0

y

′′

=0

Since

�

−

2

�

e

−2x

is always positive

(

4

�

−

4

)

=

0

(4x−4)=0

�

=

1

x=1

We need to check sign of

�

′

′

y

′′

for x < 1 and x >1

Explanation:

�

′

′

=

�

−

2

⋅

0

(

4

⋅

0

−

4

)

=

1

(

0

−

4

)

=

−

4

<

0

y

′′

=e

−2⋅0

(4⋅0−4)=1(0−4)=−4<0 ................concave downward

when x > 1 (let x=2)

�

′

′

=

�

−

2

⋅

2

(

4

⋅

2

−

4

)

=

�

−

4

⋅

4

≈

0.07

>

0

y

′′

=e

−2⋅2

(4⋅2−4)=e

−4

⋅4≈0.07>0 ............concave upward

Thus, graph of

�

=

�

�

−

2

�

y=xe

−2x

is concave down for x <1

In interval notation, graph of

�

=

�

�

−

2

�

y=xe

−2x

is concave down in

(

−

∞

,

1

)

(−∞,1