1 answer

How do you determine whether the function #F(x)= 1/12X^4 + 1/6X^3-3X^2-2X+1# is concave up or concave down and its intervals?

Question:

How do you determine whether the function #F(x)= 1/12X^4 + 1/6X^3-3X^2-2X+1# is concave up or concave down and its intervals?

Answers

To find intervals on which the graph of #F# is concave up and those on which it is concave down, investigate the sign of the second derivative.

Explanation:

For, #F(x) = 1/12x^4+1/6x^3-3x^2-2x+1# we have

#F'(x) = 1/3x^3+1/2x^2-6x-2# and

#F''(x) = x^2+x-6 = (x+3)(x-2)#

The only chance #F''(x)# has to (perhaps) change sign is at #F''(x) = 0#.

Which happens at #x=-3# and at #x=2#

On #(-oo,-3)#, both factor of #F''# are negative, so #F''# is positive and the graph of #f# is concave up.

On #(-3,2)#, we get #F''(x)# is negative, so the graph is concave down.

On #(2,oo)#, #F''(x)# is positive, so the graph is concave up.

The graph of #F# is concave up on #(-oo,-3)# and on #(2,oo)# and it is concave down on #(-3,2)#

.

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