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

Question

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

Evelyn Agard · Accepted Answer

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

Cameron Alexander · Answer

undefined To determine whether the function $ f(x) = (2x - 1)^2(x - 3)^2 $ is concave up or concave down, you can examine the sign of its second derivative.

1. Find the first derivative of $ f(x) $ using the product rule.
$$ f'(x) = 2(2x - 1)(x - 3)^2 + 2(x - 3)(2x - 1)^2 $$

2. Simplify the first derivative.

3. Find the second derivative of $ f(x) $ by differentiating $ f'(x) $.

4. Determine the sign of the second derivative for critical points. If the second derivative is positive, the function is concave up in that region. If it is negative, the function is concave down in that region.

5. Locate any critical points by setting the second derivative equal to zero and solving for $ x $.

6. Determine the intervals where the second derivative is positive or negative, indicating concave up or concave down, respectively.