For what values of x is f(x)=−x3+x2+2x+5 concave or convex?

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szalbierzfytg · Accepted Answer

Step 1To find where the equation is concave and convex we need to first find the first derivative using the power rule.ddxxn=nxn−1The first derivative is:f′(x)=−3x2+2x+2Now we find the second derivative:f″(x)=−6x+2We set it equal to zero and solve:−6x+2=0x=13Now we create a test interval from (−∞, 13) and (13, ∞)Then we pick a number from the left and right and plug it into oursecond derivative f″(x)=−6x+2Keep in mind that it can be any number I picked 0 which is a number from the left of13, then I selected 1 which is a number from the right of 13Now if the number you get after plugging it in is positive then it means for that interval the graph is concave up. If the number is negative then the graph is concave down for that interval only.So we can say that for f″(0)=2 the graph is concave up.Now for f″(1)=−4 the graph is concave down.f(x) is concave up from (−∞, 13)f(x) is concave down from (13, ∞)

For what values of x is \(\displaystyle{f{{\left({x}\right)}}}=-{x}^{{{3}}}+{x}^{{{2}}}+{2}{x}+{5}\)

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