Find values of constants so two polynomials intersect with the same slope at a point

Let $f$ and $g$ be the polynomials:

$f(x) = x^2 + ax + b, \qquad g(x) = x^3 - c.$

Find values of $a,b,c$ such that $f$ and $g$ intersect at the point $(1,2)$ and have the same tangent line at this point.

Since we want $f$ and $g$ to intersect at the point $(1,2)$ we must have $f(1) = g(1) = 2$ . This gives us the equations,

$f(1) = 2 \quad \implies \quad 1 + a + b = 2,$

and,

$g(1) = 2 \quad \implies \quad 1 - c = 2 \quad \implies \quad c = -1.$

Next, we compute the derivatives so that we can find the slope of the tangent lines at this point,

$f'(x) = 2x + a, \qquad g'(x) = 3x^2.$

Since these must be the same at the point $(1,2)$ we have

$f'(1) = g'(1) \quad \implies \quad 2 + a = 3 \quad \implies \quad a = 1.$

Then from above we know $1+a+b = 2$ ; thus, $b = 0$ .

Therefore, the values of the constants are

$a = 1, \qquad b = 0, \qquad c = -1.$

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