Is #f(x)=-x^5-21x^4-2x^3+4x-30# concave or convex at #x=0#?

Question

Evelyn Agard · Accepted Answer

inflexion point

Calculating the first derivative of #f(x)=-x^5-21x^4-2x^3+4x-30# at #x=0# we get #f_x(0)=0# The second derivative gives #f_{x x}(0) = 0# and the third #f_{x x x}(0)=-12# so the origin is an inflexion point. In this point the function behavior is like #-abs(c_0) x^3#. Nor concave or convex.

Aurora Alvarado · Answer

undefined To determine the concavity or convexity of a function at a point, we need to examine the second derivative of the function at that point.

Given $ f(x) = -x^5 - 21x^4 - 2x^3 + 4x - 30 $, the second derivative is $ f''(x) = -60x^2 - 84x - 6 $.

To find the concavity or convexity at $ x = 0 $, we evaluate $ f''(0) $:
$$ f''(0) = -60(0)^2 - 84(0) - 6 = -6 $$

Since the second derivative is negative at $ x = 0 $, the function is concave at $ x = 0 $.

Luke Alvarez · Answer

undefined To determine if $ f(x) = -x^5 - 21x^4 - 2x^3 + 4x - 30 $ is concave or convex at $ x = 0 $, we need to examine the second derivative of the function at that point.

First, find the first derivative of $ f(x) $, then evaluate the second derivative at $ x = 0 $.

$$ f'(x) = -5x^4 - 84x^3 - 6x^2 + 4 $$

Now, find the second derivative:

$$ f''(x) = -20x^3 - 252x^2 - 12x $$

Evaluate the second derivative at $ x = 0 $:

$$ f''(0) = 0 $$

Since the second derivative at $ x = 0 $ is zero, the concavity of the function cannot be determined solely from this point. Additional information or analysis of the function's behavior around $ x = 0 $ is needed to determine if it is concave, convex, or neither at that point.