For what values of x is #f(x)=(7x-1)(x-6)(x-2)# concave or convex?

Question

Ezra Adams · Accepted Answer

#f(x)=(7x-1)(x-6)(x-2)# is concave down on the interval #x<19/7#, and #f(x)# is concave up on the interval #x>19/7#.

#f(x)=(7x-1)(x-6)(x-2)# Find first derivative: #f'(x)=21x^2-114x+92# Find second derivative: #f''(x)=42x-114# #f''(x)=6(7x-19)# Find where second derivative is negative to get intervals of concave down, and where #f''(x)# is positive is where #f(x)# is concave up. #f''(x)# is positive when #x>19/7#, and #f''(x)# is negative when #x<19/7#.

Evelyn Agard · Answer

undefined To determine the concavity or convexity of the function $ f(x) = (7x - 1)(x - 6)(x - 2) $, you need to find the intervals where its second derivative is positive (convex) or negative (concave).

First, find the second derivative of $ f(x) $, denoted as $ f''(x) $. Then, identify the critical points by solving for $ f''(x) = 0 $ and analyze the sign of $ f''(x) $ in the intervals between these critical points.

To summarize:
1. Find $ f''(x) $.
2. Determine critical points by solving $ f''(x) = 0 $.
3. Analyze the sign of $ f''(x) $ in each interval defined by the critical points.

Once you determine the intervals where $ f(x) $ is concave or convex, you can state the values of $ x $ within those intervals.