Is #f(x)=(x-3)^3-x+15# concave or convex at #x=3#?

Question

Scarlett Allison · Accepted Answer

Neither. #x= 3# is an inflection point.

You have to find the second derivative to answer this question. I would use the chain rule to differentiate #(x- 3)^3#. Let #u = x - 3# and #y = u^3#. Then #y' = 3u^2 * u' = 3u^2 * 1 = 3(x - 3)^2# #f'(x) = 3(x- 3)^2 - 1# Now it's easy enough to expand to find the second derivative using the power rule. #f'(x) = 3(x^2 - 6x - 9) - 1# #f'(x) = 3x^2 - 18x - 27 - 1# #f'(x) = 3x^2 - 18x - 28# Now find the second derivative. #f''(x) = 6x - 18# We now test the sign of the second derivative at #x = 3# under the knowledge that if: #•f''(x) > 0#, at #x = a# then #f(x)# is concave at #x = a# #•f''(x) < 0#, at #x = a# then #f(x)# is convex at #x = a# We have: #f''(3) = 6(3) - 18 = 18 - 18 = 0# So #x = 3# is a point of inflection, so the graph is neither concave up nor concave down at that point. Instead its changing between the two. Hopefully this helps!

Evelyn Agard · Answer

undefined To determine whether $ f(x) = (x - 3)^3 - x + 15 $ is concave or convex at $ x = 3 $, we need to examine the second derivative of the function at that point.

1. Find the first derivative of $ f(x) $:
$$ f'(x) = 3(x - 3)^2 - 1 $$

2. Find the second derivative of $ f(x) $:
$$ f''(x) = 6(x - 3) $$

3. Evaluate the second derivative at $ x = 3 $:
$$ f''(3) = 6(3 - 3) = 0 $$

Since the second derivative at $ x = 3 $ is equal to zero, the function neither concave nor convex at that point. Instead, it exhibits an inflection point.

Charles Anctil · Answer

undefined To determine whether the function $ f(x) = (x - 3)^3 - x + 15 $ is concave or convex at $ x = 3 $, we can analyze the sign of the second derivative of the function at that point.

1. Find the first derivative of $ f(x) $:
$$ f'(x) = 3(x - 3)^2 - 1 $$

2. Find the second derivative of $ f(x) $:
$$ f''(x) = 6(x - 3) $$

3. Evaluate the second derivative at $ x = 3 $:
$$ f''(3) = 6(3 - 3) = 0 $$

Since the second derivative $ f''(3) $ is equal to zero, we cannot determine the concavity or convexity of the function at $ x = 3 $ using this method. Further analysis or another method may be needed to determine the concavity or convexity at that point.