Answered

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The second derivative of the function f
f
is given by f′′(x)=x2cos(x√)−2xcos(x√)+cos(x√)
f
″
(
x
)
=
x
2
⁢
cos
⁡
(
x
)
−
2
x
⁢
cos
⁡
(
x
)
+
cos
⁡
(
x
)
. At what values of x
x
in the interval (0,3)
(
0
,
3
)
does the graph of f
f
have a point of inflection?

2.467 only
Answer A: 2.467 only
A

1 and 2.467
Answer B: 1 and 2.467
B

1.443 and 2.734
Answer C: 1.443 and 2.734
C

1 and 1.962

Sagot :

According to it's second derivative, the function has points of inflection given by:

Answer B: 1 and 2.467.

What are the points of inflection of a function?

The points of inflection of a function are the values of x for which:

[tex]f^{\prime\prime}(x) = 0[/tex]

In this problem, the second derivative is:

[tex]f^{\prime\prime}(x) = x^2\cos{(\sqrt{x})} - 2x\cos{(\sqrt{x})} +\cos{(\sqrt{x})}[/tex]

Hence:

[tex]x^2\cos{(\sqrt{x})} - 2x\cos{(\sqrt{x})} +\cos{(\sqrt{x})} = 0[/tex]

[tex](x^2 - 2x + 1)\cos{(\sqrt{x})} = 0[/tex]

[tex]x^2 - 2x + 1 = 0 \rightarrow (x - 1)^2 = 0 \rightarrow x = 1[/tex]

[tex]\cos{(\sqrt{x})} = 0[/tex]

[tex]\sqrt{x} = \frac{\pi}{2}[/tex]

[tex](\sqrt{x})^2 = \left(\frac{\pi}{2}\right)^2[/tex]

[tex]x = 2.467[/tex]

Hence option B is correct.

You can learn more about points of inflection at https://brainly.com/question/10352137

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