What is the inflection 4x+8cos(x)(0.2pi) ?

The inflection 4x+8cos(x)(0.2pi) is

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inflection 4x+8cos(x)(0.2pi)

inflection points

4 x + 8 cos (x) (0.2 π)

(\frac{π}{2} + 2 πn, 4 (\frac{π}{2} + 6.28318 \dots n) + 5.02654 \dots cos (\frac{π}{2} + 6.28318 \dots n)), (\frac{3 π}{2} + 2 πn, 4 (4.71238 \dots + 6.28318 \dots n) + 5.02654 \dots cos (4.71238 \dots + 6.28318 \dots n))

Solution steps

Find where

f^{''} (x)

is equal to zero or undefined

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Identify inflection points not in domain or where

f (x)

is not continuous

Domain of

4 x + 8 cos (x) 0.2 π : - \infty < x < \infty

All inflection points are in domain and not on domain edges

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Find intervals:

Concave Downward

: 2 πn \leq x < \frac{π}{2} + 2 πn,

Concave Upward

: \frac{π}{2} + 2 πn < x < \frac{3 π}{2} + 2 πn,

Concave Downward

: \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Plug

x = \frac{π}{2} + 2 πn

into

4 x + 8 cos (x) 0.2 π : 4 (\frac{π}{2} + 6.28318 \dots n) + 5.02654 \dots cos (\frac{π}{2} + 6.28318 \dots n)

(\frac{π}{2} + 2 πn, 4 (\frac{π}{2} + 6.28318 \dots n) + 5.02654 \dots cos (\frac{π}{2} + 6.28318 \dots n))

Plug

x = \frac{3 π}{2} + 2 πn

into

4 x + 8 cos (x) 0.2 π : 4 (4.71238 \dots + 6.28318 \dots n) + 5.02654 \dots cos (4.71238 \dots + 6.28318 \dots n)

(\frac{3 π}{2} + 2 πn, 4 (4.71238 \dots + 6.28318 \dots n) + 5.02654 \dots cos (4.71238 \dots + 6.28318 \dots n))

(\frac{π}{2} + 2 πn, 4 (\frac{π}{2} + 6.28318 \dots n) + 5.02654 \dots cos (\frac{π}{2} + 6.28318 \dots n)), (\frac{3 π}{2} + 2 πn, 4 (4.71238 \dots + 6.28318 \dots n) + 5.02654 \dots cos (4.71238 \dots + 6.28318 \dots n))

f (x) = 5^{x^{2} - 4 x - 2}

f (x) = 2 x^{3} - 2 x^{2} - 32 x + 32

(\frac{1}{x})^{2} - 3 (\frac{1}{x})

y = \frac{2}{3} x

\frac{x - 1}{2}