How do you determine whether the function f(x)=2x−3x2 is concave up or concave down and its intervals?

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gigglesbuggk1co · Accepted Answer

f(x)=2x−3x2 is concave up when x&amp;gt;92 and concave down when x&amp;lt;92 (and x≠0).Explanation:Step 1By the Quotient Rule, for x≠0, the first derivative isf′(x)=x2⋅2−(2x−3)⋅2xx4=−2x2+6xx4=−2x+6x3and the second derivative isfx)=x3⋅(−2)−(−2x+6)⋅3x2x6=4x3−18x2x6=4x−18x4Step 2Since x4≥0 for all x, the sign of f&quot;(x) is the same as the sign of its numerator 4x−18. This expression is positive when 4x&amp;gt;18⇔x&amp;gt;92 and negative when 4x&amp;lt;18⇔x&amp;lt;92.The value x=92 is the first coordinate of the unique inflection point of the graph of f. The second coordinate is f(92)=9−3(92)2=6814=2481=827.

How do you determine whether the function

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