How do you determine whether the function f(x)=2x3−3x2−12x+1 is concave up or concave down and its intervals?

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blessgansgxei · Accepted Answer

Answer:Investigate the sign of the second derivative.f is concave down on (−∞,12) and concave up on (12,∞).(12,−112) is the inflection point for the graph of f.Step 1f(x)=2x3−3x2−12x+1f′(x)=6x2−6x−12f″(x)=12x−6f&quot;(x) is never undefined and is 0 only at x=12So the only x value at which the concavity might change is 12.For x&amp;lt;12 we find that fx)&amp;lt;0 and for x&amp;gt;12,fx)&amp;gt;0Step 2So the graph of f is concave down on (−∞,12) and concave up on (12,∞).f(12)=2(12)3−3(12)2−12(12)+1=−112(12,−112) is the only inflection point for the graph of f.

How do you determine whether the function

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