What is the tangent of y=-x^3+2x^2+1 ?

The tangent of y=-x^3+2x^2+1 is y=(-3a_{0}^2+4a_{0})x+2a_{0}^3-2a_{0}^2+1

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tangent of y=-x^3+2x^2+1

tangent of

y = - x^{3} + 2 x^{2} + 1

y = (- 3 a_{0}^{2} + 4 a_{0}) x + 2 a_{0}^{3} - 2 a_{0}^{2} + 1

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, - a_{0}^{3} + 2 a_{0}^{2} + 1)

Find the slope of

y = - x^{3} + 2 x^{2} + 1 : \frac{d y}{d x} = - 3 x^{2} + 4 x

EN : T i tl e G e n er a l Eq u a t i o n Sl o p e A t P o in t 2 Eq : m = - 3 a_{0}^{2} + 4 a_{0}

Find the line with slope m=

- 3 a_{0}^{2} + 4 a_{0}

and passing through

(a_{0}, - a_{0}^{3} + 2 a_{0}^{2} + 1) : y = (- 3 a_{0}^{2} + 4 a_{0}) x + 2 a_{0}^{3} - 2 a_{0}^{2} + 1

y = (- 3 a_{0}^{2} + 4 a_{0}) x + 2 a_{0}^{3} - 2 a_{0}^{2} + 1

What is the tangent of y=-x^3+2x^2+1 ?
The tangent of y=-x^3+2x^2+1 is y=(-3a_{0}^2+4a_{0})x+2a_{0}^3-2a_{0}^2+1