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Find the points of local maxima and local minima of $f(x)=x^3-6x^2+9x+2014, x\in R$.

  • What is local maxima & local minima?
  • How do get the points of local maxima & local minima of a function?

Really appreciate your help. Thanks.

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    $\begingroup$ Points where the derivative is zero. Do you know how to differentiate a polynomial? $\endgroup$
    – Vishu
    Apr 12, 2020 at 17:59
  • $\begingroup$ I’m pretty sure that this is all covered in excruciating detail in the course material that preceded this exercise that you’re now attempting. $\endgroup$
    – amd
    Apr 12, 2020 at 23:53

1 Answer 1

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A local maximizer is a point where a function rises to a value (the local maximum value) and then decreases again. A local minimizer is a point where a function falls to a value (the local minimum value) and then rises again.

Where continuous functions have a maximum or minumum, a tangent line to the function will be horizontal, i.e. have a slope of zero. This means the derivative there will be zero. Points where $f'(x)=0$ are called critical points.

To check if these points are local minimums, maximums, or inflection points, look at the second derivative. If $f''(x)>0$, then $x$ is a minimizer since $f$ is concave up. If $f''(x)<0$, then $x$ is a maximizer since $f$ is concave down. Points where $f''(x)=0$ are called inflection points, and indicate a potential change in concavity.

Concavity indicates the overall shape of a function. Concave up means it is shaped like part of a bowl that is right-side up. Concave down means it is shaped like part of a bowl that is upside-down. See the below figure for a visual example.

visual example

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