For what values of x is f(x)=(x−2)(x−7)(x−3) concave or convex?

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Dubliddissefyf8 · Accepted Answer

Step 1The concavity and convexity of a function are determined by the sign (positive/negative) of the second derivative.If f″(a)&amp;lt;0, then f(x) is concave at x=aIf f″(a)&amp;gt;0, then f(x) is convex at x=aIn order to find the second derivative, we should first simplify the undifferentiated function by distributing.f(x)=(x2−9x+14)(x−3)=x3−12x2+41x−42Now, find the first and second derivatives through a simple application of the power rule.f′(x)=3x2−24x+41f″(x)=6x−24Now, we must find the times when 6x−24 is positive and when it is negative. The times when the function could shift from positive to negative or vice versa are when6x−24=06x−24=06x=24x=4The sign of the second derivative, and by extension, the concavity/convexity of the function, could shift only at x=4. Thus, we should test points on either side of x=4 to determine which concavity/convexity is present on either side.When x&amp;lt;4:Test point at x=0f″(0)=6(0)−24=−24Since this is &amp;lt;0, the entire interval (−∞, 4) will be concave.When x&amp;gt;4:Test point at x=5:f″(5)=6(5)−24=6Since this is &amp;gt;0, the entire interval (4, +∞) will be convex.We can check the graph of f(x): convexity is characterized by a ∪ shape, and concavity is characterized by a ∩ shape. The concavity of the graph should shift at x=4

For what values of x is \(\displaystyle{f{{\left({x}\right)}}}={\left({x}-{2}\right)}{\left({x}-{7}\right)}{\left({x}-{3}\right)}\)

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