Find all relative extrema and points of inflection for the function 
. Be sure to give coordinate pairs for each point. You do not need to draw the graph.
| Foundations:
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| Since our function is a polynomial, the relative extrema occur when the first derivative is zero. We then have two choices for finding if it is a local maximum or minimum:
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Second Derivative Test: If the first derivative at a point  is  , and the second derivative is negative (indicating it is concave-down, like an upside-down parabola), then the point  is a local maximum.
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| On the other hand, if the second derivative is positive, the point is a local minimum. You can also use the first derivative test, but it is usually a bit more work! For inflection points, we need to find when the second derivative is zero, as well as check that the second derivative "splits" on both sides.
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Solution:
| Step 1:
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| Find the first and second derivatives: Based on our function, we have
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| Similarly, from the first derivative we find
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| Step 2:
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| Find the roots of the derivatives: We can rewrite the first derivative as
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from which it should be clear we have roots  and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -3}
.
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| On the other hand, for the second derivative, we have
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g\,''(x)\,=\,4x+2\,=\,2\left(x+{\frac {1}{2}}\right).}
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This has a single root:  .
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| Step 3:
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| Test the potential extrema: We know that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2,-3}
are the candidates. We check the second derivative, finding
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- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\,''(2)\,=\,4\cdot 2+2\,>\,0,}
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| while
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- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\,''(-3)\,=\,2(-3)+2\,<\,0.}
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| Note that
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- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(2)\,=\,\frac{2}{3}(8)+4-24\,=\,-\frac{44}{3},}
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| while
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- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(-3)\,=\,\frac{2}{3}(-27)+9-12(-3)\,=\,27.}
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| By the second derivative test, the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2,g(2))=\left(2,-\frac{44}{3}\right)}
is a relative minimum, while the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-3,g(-3))=(-3,27)}
is a relative maximum.
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| Step 4:
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Test the potential inflection point: We know that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\,''(-1/2)=0}
. On the other hand, it should be clear that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<-1/2}
, then  . Similarly, if  , then  . Thus, the second derivative "splits" around  (i.e., changes sign), so the point  is an inflection point.
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| Since
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g\left(-{\frac {1}{2}}\right)\,=\,{\frac {2}{3}}\cdot -{\frac {1}{8}}+{\frac {1}{4}}-12\left(-{\frac {1}{2}}\right)\,=\,{\frac {19}{4}},}
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| our inflection point is
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left(-{\frac {1}{2}},{\frac {19}{4}}\right).}
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| Final Answer:
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There is a local minimum at  , a local maximum at  and an inflection point at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left(-{\frac {1}{2}},{\frac {19}{4}}\right).}
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