How do you determine whether the function #f(x)=(2x-1)^2(x-3)^2# is concave up or concave down?

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Answer 1 · 2015-09-11T16:09:23

To determine where the graph of #f# is concave up and where it is concave down, look at the sign of #f''(x)#.

#f(x)=(2x-1)^2(x-3)^2#

#f'(x) = 2(2x-1)*(2)(x-3)^2 + (2x-1)^2 2(x-3)(1)#

# = 2(2x-1)(x-3)[2(x-3)+(2x-1)]#

# = 2(2x-1)(x-3)(4x-7)#

#f''(x) = 2(2)(x-3)(4x-7)#

# + 2(2x-1)(1)(4x-7)#

#+2(2x-1)(x-3)(4)#

# = 48x^2-168x+61#

Set #f''(x) = 0# to find partition numbers: #(21 +- 5sqrt3)/12#

Test each interval:

On #(-oo, (21 - 5sqrt3)/12)#, #f''# is positive, graph is concave up

on #(21 - 5sqrt3)/12,(21 + 5sqrt3)/12#, #f''# is negative, graph is concave down

on #((21 + 5sqrt3)/12,oo)#, #f''# is positive, graph is concave up