As D approaches zero

Calculus Level 5

$\large f_k(x) = A(k)x^2 + B(k)x + C(k),$ Let $k$ be a real parameter and $A$ , $B$ and $C$ are real, continuous functions on $\mathbb R$ .

Suppose that for a certain interval $k \in \langle p, q \rangle$ , the equation $f_k(x) = 0$ has two real solutions; but at the end of the interval, $f_q(x) = 0$ does not have two real solutions.

What more can you say about the solutions of $f_q(x) = 0$ ?

The equation must have one real solution. This equation can have zero, one, or infintely many real solutions. The equation must have no real solution.

1 solution

Arjen Vreugdenhil
Apr 9, 2016

The number of solutions of the equation $Ax^2 + Bx + C$ generally depends on the discriminant, $D = B^2 - 4AC.$ Since $D$ is a continuous function of $A$ , $B$ , and $C$ , which in turn are continuous functions of $k$ , the discriminant is a continuous function of the parameter $k$ .

If on the interval $\langle p, q\rangle$ the equation has two solutions, we must have $D(k) > 0$ on that interval, and $\lim_{k \to q} D(k) \geq 0$ as well. But there are less than two solutions at $q$ , so that $D(q) = 0$ . This would suggest that $f_q(x) = 0$ has only one solution.

However, there is one exception: if $A(q) = 0$ , then the function $f_q$ is not quadratic but linear. Then there is

-- one zero if $B(q) \not = 0$ ;

-- no zeroes if $B(q) = 0$ and $C(q) \not = 0$ ;

-- infinitely many zeroes if $B(q) = C(q) = 0$ .

As D approaches zero

1 solution

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