f(x) = - x² + ax + b ← this is the parábola

The line is tangent to the curve. That means that there is only one common point between the line and the parabola.

y = x + 1 ← this is the line (ℓ1)

y = - x² + ax + b

y = y

x + 1 = - x² + ax + b

x + 1 + x² - ax - b = 0

x² + x.(1 - a) + (1 - b) = 0

Polynomial like: ax² + bx + c, where:

A = 1

B = (1 - a)

C = (1 - b)

Δ = B² - 4AC (discriminant)

Δ = (1 - a)² - 4.(1 - b)

Δ = 1 - 2a + a² - 4 + 4b

Δ = a² - 2a + 4b - 3 → only one point, → Δ = 0

a² - 2a + 4b - 3 = 0 ← first condition → equation (1)

y = - 2x + 1 ← this is the line (ℓ2)

y = - x² + ax + b

y = y

- 2x + 1 = - x² + ax + b

- 2x + 1 + x² - ax - b = 0

x² + x.(- 2 - a) + (1 - b) = 0

Polynomial like: ax² + bx + c, where:

A = 1

B = (- 2 - a)

C = (1 - b)

Δ = B² - 4AC (discriminant)

Δ = (- 2 - a)² - 4.(1 - b)

Δ = 4 + 4a + a² - 4 + 4b

Δ = a² + 4a + 4b → only one point, → Δ = 0

a² + 4a + 4b = 0 ← second condition → equation (2)

Then you calciulate (2) - (1)

(a² + 4a + 4b) - (a² - 2a + 4b - 3) = 0 - 0

a² + 4a + 4b - a² + 2a - 4b + 3 = 0

6a + 3 = 0

6a = - 3

a = - 3/6

→ a = - 1/2

Recall (2): a² + 4a + 4b = 0

(- 1/2)² + [4 * (- 1/2)] + 4b = 0

(1/4) - 2 + 4b = 0

4b = 2 - (1/4)

4b = 7/4

→ b = 7/16

f(x) = - x² + ax + b

f(x) = - x² - (1/2).x + (7/16)

Hola

Recta tangente en punto (xo,yo)

yo = xo^2 + a xo + b

y - yo = (2 xo + a) (x - xo)

y = xo^2 + a xo + b - (2 xo^2 + a xo) + (2 xo + a) x

y = (2 xo + a) x + (b - xo^2)

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Rectas tangentes

Primera recta tangente

y = x + 1

1) 2 xo + a = 1

2) b - xo^2 = 1

de 1)

xo = (1 - a)/2

en 2)

b - ((1 - a)/2)^2 = 1

3) 4 b - (1 - a)^2 = 4

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Segunda recta tangente

y = -2 x + 1

4) 2 x1 + a = -2

5) b - x1^2 = 1

de 4)

x1 = (-2 - a)/2

en 5)

b - ((-2 - a)/2)^2 = 1

6) 4 b - (2 + a)^2 = 4

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de 3) y 6)

(1 - a)^2 = (2 + a)^2

1 - 2 a + a^2 = 4 + 4 a + a^2

6 a = -3

a = -1/2

**********

en 6)

4 b - (2 - (1/2))^2 = 4

4 b - (3/2)^2 = 4

4 b = 4 + (9/4)

4 b = 25/4

b = 25/16

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parábola

y = x^2 - (1/2) x + (25/16)

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Saludos

consulta al profesor poztnagel