Is #f(x)=4x^5-x^4-9x^3+5x^2-7x# concave or convex at #x=-1#?

Question

Cameron Alexander · Accepted Answer

concave at x = -1

To determine if a function is concave/convex we calculate the value of f''(x) at x = a. • If f''(a) > 0 , then f(x) is convex at x = a • If f''(a) < 0 , then f(x) is concave at x = a #f(x)=4x^5-x^4-9x^3+5x^2-7x# differentiate using the#color(blue)" power rule"# #rArrf'(x)=20x^4-4x^3-27x^2+10x-7# and #f''(x)=80x^3-12x^2-54x+10# so #f''(-1)=80(-1)^3-12(-1)^2-54(-1)+10# #=-80-12+54+10=-28# Since f''(-1) < 0 , then f(x) is concave at x = -1 graph{4x^5-x^4-9x^3+5x^2-7x [-36.52, 36.53, -18.24, 18.29]}

Adam Adkins · Answer

undefined To determine the concavity of the function $ f(x) = 4x^5 - x^4 - 9x^3 + 5x^2 - 7x $ at $ x = -1 $, we need to analyze the second derivative of the function at that point.

First, find the second derivative of $ f(x) $:
$$ f'(x) = 20x^4 - 4x^3 - 27x^2 + 10x - 7 $$
$$ f''(x) = 80x^3 - 12x^2 - 54x + 10 $$

Now, evaluate $ f''(-1) $:
$$ f''(-1) = 80(-1)^3 - 12(-1)^2 - 54(-1) + 10 $$
$$ f''(-1) = -80 - 12 + 54 + 10 $$
$$ f''(-1) = -28 $$

Since $ f''(-1) = -28 $ is negative, the function $ f(x) $ is concave downward at $ x = -1 $.