The polynomial function f is defined by f(x)=-3x^4+7x^3+3x^2-9x-1.
Find all the points (x,f(x)) where there is a local minimum.

Round to the nearest hundredth.
If there is more than one point, enter them using the "and" button.

Question

The polynomial function f is defined by f(x)=-3x^4+7x^3+3x^2-9x-1. 
 Find all the points (x,f(x))  where there is a local minimum. 

 Round to the nearest hundredth. 
 If there is more than one point, enter them using the "and" button.

ganeshie8 · Answer

calculus ?

anonymous · Answer

yes

ganeshie8 · Answer

first step is to find the "critical points"

ganeshie8 · Answer

knw how to find them ?

anonymous · Answer

no. first problem I'm encountering like this

anonymous · Answer

So first find critical points by taking the derivative and setting it equal to 0 or Undefined. In this case you only have zeros.

$$f \prime=-12x^3+21x^2+6x-9$$
so set that equal to 0. I'm assuming you are able to use a calculator or wolfram alpha to solve for the x values.

then once you have the critical points, you can either use the second derivative test or individually test the points/look at a graph. I'm assuming you haven't been taught the second derivative test yet so I'd recommend looking at a graph. If you want me to explain the second derivative test though, let me know!

anonymous · Answer

yes, please

anonymous · Answer

so second derivative test, as the name implies, involves taking the second derivative and looking at your critical points. If the second derivative evaluates as positive, then the function is concave up at that critical point and therefore the point is a local minimum. if the second derivative is negative, the function is concave down and therefore the point must be a maximum. If the second derivative at that point is =0, then the test is inconclusive as the function changes concavity at that point. |dw:1393392032096:dw|