Find the points of inflection of the graph of the function. f(x) = 7 sin x + sin 2x, [0, 2\pi]...

Question:

Find the points of inflection of the graph of the function.

f(x) = 7 sin x + sin 2x, {eq}[0, 2\pi] {/eq}

(x, y)= ( ) (smallest x-value)

(x, y)= ( ) (largest x-value)

Inflection point

A function with one independent variable has an inflection point where its second derivative is zero, therefore an inflection point is found solving the second derivative expression, or with the second derivative graph, identifying the x-axis interceptions.

Answer and Explanation:


The function is:


{eq}\displaystyle \ f(x) = 7 sin x + sin 2x \\ {/eq}


The first derivative is:


{eq}\displaystyle \ f'(x)= 7\,\cos \left( x \right) +2\,\cos \left( 2\,x \right) \\ \displaystyle \ f'(x)= 7\,\cos \left( x \right) +4\, \left( \cos \left( x \right) \right) ^{2}-2 \\ \displaystyle \ f'(x)= \left( \cos \left( x \right) +2 \right) \left( 4\,\cos \left( x \right) -1 \right) \\ {/eq}


The second derivative is:


{eq}\displaystyle \ f''(x) = -7\,\sin \left( x \right) -4\,\sin \left( 2\,x \right) \\ {/eq}

Inflection point exist where {eq}\ f''(x) =0 {/eq} so,

Thun, the inflection points are:

{eq}\displaystyle 0 = \ f''(x) \; \Rightarrow \; 0= -7\,\sin \left( x \right) -4\,\sin \left( 2\,x \right) \\ {/eq}

With algebraic manipulation is impossible to find the roots or inflection points of this equation, but we can graph the second derivative and see its x-axis interceptions.


The second derivative and its x-axis interceptions


With close up:


first inflection point at x=0


Second inflection point at x= 2.637


Third inflection point at x= 3.142


Fourth inflection point at x= 3.636


Last inflection point at x=6.283


Conclusion:


The inflection points are

{eq}\displaystyle (0,f(0)) \; \Rightarrow \; (0, 0) \\ \displaystyle ( 2.637,f(2.637)) \; \Rightarrow \; (2.637, 2.537758046) \\ \displaystyle ( 3.142,f(3.142)) \; \Rightarrow \; (3.142, - 0.002036732062) \\ \displaystyle ( 3.636,f(3.636)) \; \Rightarrow \; (3.636, - 2.486195334) \\ \displaystyle ( 6.283,f(6.283)) \; \Rightarrow \; (6.283, - 0.001667764601) \\ {/eq}


The function and its second derivative


Learn more about this topic:

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Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5
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