Is #f(x)=x^4-4x^3+x-4# concave or convex at #x=-1#?

Question

Ezra Adams · Accepted Answer

convex

To find that out, we need to get the second derivative first. Getting the first derivative. #[1]" "f'(x)=d/dx(x^4-4x^3+x-4)# We can easily find this using power rule. #[2]" "f'(x)=4x^3-12x^2+1# Getting the second derivative. #[1]" "f''(x)=d/dx(4x^3-12x^2+1)# Use power rule again. #[2]" "f''(x)=12x^2-24x# Now that we know the second derivative, we will evaluate #f''(x)# at #x=-1# to check its concavity. • If #f''(x)>0#, then it is concave up or convex • If #f''(x)<0#, then it is concave down or concave #[1]" "f''(-1)=12(-1)^2-24(-1)# #[2]" "f''(-1)=12+24# #[3]" "f''(-1)=36# Since #f(x)# is 36 at #x=-1# and 36 is greater than 0, then #f(x)# is convex at #x=-1#.

Axel Alvarez · Answer

undefined To determine whether $ f(x) = x^4 - 4x^3 + x - 4 $ is concave or convex at $ x = -1 $, we need to analyze the second derivative of the function at that point.

The second derivative of $ f(x) $ is denoted as $ f''(x) $, and it provides information about the concavity of the function.

First, we find the first derivative of $ f(x) $:
$$ f'(x) = 4x^3 - 12x^2 + 1 $$

Now, we find the second derivative:
$$ f''(x) = 12x^2 - 24x $$

To determine the concavity at $ x = -1 $, we plug $ x = -1 $ into $ f''(x) $:
$$ f''(-1) = 12(-1)^2 - 24(-1) = 12 - (-24) = 12 + 24 = 36 $$

Since $ f''(-1) = 36 $ and $ 36 > 0 $, the function is concave upward at $ x = -1 $.