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

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?

• Points where the derivative is zero. Do you know how to differentiate a polynomial? Apr 12, 2020 at 17:59
• 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.
– amd
Apr 12, 2020 at 23:53

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.