Question

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

Answer 1

concave at x = -1

Explanation:

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]}