The function f (x) = - x ^ 3 + ax ^ 2 + BX + C is known. The tangent equation at P (1, - 2) on the image is y = - 3 + 1 If the function f (x) increases monotonically in the interval [- 2,0], the value range of real number B is obtained | Math enthusiast | Mathematical art

The function f (x) = - x ^ 3 + ax ^ 2 + BX + C is known. The tangent equation at P (1, - 2) on the image is y = - 3 + 1 If the function f (x) increases monotonically in the interval [- 2,0], the value range of real number B is obtained

Derivation
f'(x)=-3x^2+2ax+b
The tangent equation at point P (1, - 2) is y = - 3x + 1
So - 3 = - 3 + 2A + B
-2=-1+a+b+c
A = - B / 2
c=-1-b/2
F '(x) = - 3x ^ 2 + 2aX + B = - 3x ^ 2-bx + b > = 0 is constant on interval [- 2,0]
Consider from quadratic function
F '(x) = - 3x ^ 2-bx + B is a parabola with an opening downward
Therefore, if f '(x) > = 0 is always true, only the endpoint is true
f'(0)=b>=0
f'(-2)=-12+3b>=0
b>=4
If you have any questions, please ask

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