If y=3x5−5x3, what are the points of inflection of the graph f (x)?

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alastrimsmnr · Accepted Answer

Answer: x=0Step 1y=3x5−5x3y′=15x4−15x2At a critical point y′=0⇒15x4−15x2=0∴15x2(x2−1)=0∴∴15x2(x−1)(x+1)=0So there are three critical points, when x=0,x=±1Step 2To determine the nature of these critical points we look at the second derivative:y′=15x4−15x2∴y″=60x3−30x∴y30x(2x2−1)So when: x=0⇒y″=0x=−1⇒y″=(−30)(2−1)=−30x=1⇒y″=(30)(2−1)=30So the nature of the critical points is as followsx=0 point oi inflectionx=−1 maximumx=1 minimum

August Moore · Accepted Answer

Step 1Points of inflection are points on the graph at which the concavity (and the sign of the second derivative) change.y=3x5−5x3y′=15x4−15x2y″=60x3−30xThe zeros of y&#039;&#039; are found by solvingy″=30x(2x2−1)=0The solutions are x=−12,0,and 12Step 2Each of these is a zero of the polynomial y&#039;&#039; with multiplicity 1, so the sign changes at each of them. (Or make a sign table, chart, diagram, whatever you&#039;ve been taught to call it.)Therefore each is the x-value of a point of inflection. To find the points, find the corresponding y-values.The points of inflection (with rationalized denominators) are: (−22,728),(0,0)(22,−728)

If y=3x^{5}-5x^{3}, what are the points of inflection of the

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