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.
2026-04-02 19:54:19.1775159659
How to find the points of local maxima and local minima of $f(x)=x^3-6x^2+9x+2014, x\in R$.
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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.