# `f(x) = 6sin(x) + cot(x), -pi<=x<=pi` Produce graphs of `f` that reveal all the important aspects of the curve. In particular, you should use graphs of `f'` and `f''`

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Expert Answers

gsarora17 | Certified Educator

`f(x)=6sin(x)+cot(x)`

See the attached graph and link(different range), f(x) is in Red color, f' is in Blue color and f'' is in Green color.

From the graph, Vertical Asymptotes at x=0, x=pi , x=-pi

f is decreasing on the intervals (-pi,-1.40) , (-0.40,0) ,(0,0.40) and(1.40,pi)

f is increasing on the intervals(-1.40,-0.40) and(0.40,1.40)

Local minimum f(-1.40) `~~` -6 , f(0.40) `~~` 4.75

Local maximum f(-0.40) `~~` -4.75 , f(1.40)`~~` 6

`f'(x)=6cos(x)-csc^2(x)`

`f''(x)=-6sin(x)+2csc^2(x)cot(x)`

From the Graph,

function is Concave up on the intervals at about (-pi,-0.80) and (0,0.80)

function is concave down on the intervals about (-0.80,0) and(0.80,pi)

Inflection points at about (-0.80,-5.25) and (0.80,5.25)

Further Reading: