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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) .)

User Maryline
by
7.6k points

1 Answer

6 votes

Answer:

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

User Ferdinand Van Wyk
by
8.9k points