Let the function y = f (x) have a third order continuous derivative in a neighborhood of x = x0. If f "(x0) = 0 and F" '(x0) is not equal to 0, ask the relation between F' (x0) and 0,

Let the function y = f (x) have a third order continuous derivative in a neighborhood of x = x0. If f "(x0) = 0 and F" '(x0) is not equal to 0, ask the relation between F' (x0) and 0,

If f '(x) = f' (x0) + F '' (x0) (x-x0) + F '' '(x0) (x-x0) ^ 2 / 2 + O (x-x0) ^ 2 = f' (x0) + F '' '(x0) (x-x0) ^ 2 / 2 + O (x-x0) ^ 2, take x → x0, then f' (x) satisfies f '(x) > = f' (x0) near x0. This probably shows that the first derivative has an extreme value at x0, but the size relationship between F '(x0) and 0 cannot be obtained

1. F (x) = 5x ^ 3-2x ^ 2 + x-3, x0 = 0.2, f (x) = x / SiNx, x0 = π / 2.3, y = x * [(8-x) ^ 1 / 3], x0 = 0, calculate the derivative of the function at the specified point

1、f'（x）=15x^2-4x+1
f(0)=1
2、f‘（x）=（sinx-xcosx）/sin^2 x
f'(π/2)=(1-0)/1=1
3、y'=(8-x)^1/3-x/3*(8-x)^-2/3
y'(0)=8^1/3-0=2