First slide

Mean value theorems

Question

If $a, b, c$ are real numbers such that $a + b + c = 0$ , then the quadratic equation $3 {ax}^{2} + 2 bx + c = 0$ has

Moderate

A

At least one root in $[0, 1]$
B

At least one root in $[1, 2]$
C

At least one root in $[- 1, 0]$
D

None of these

Solution

Let $f^{1} (x)$ denotes the quadratic expression $f^{1} (x) = 3 {ax}^{2} + 2 bx + c,$ whose antiderivative be denoted by $f (x) = {ax}^{3} + {bx}^{2} + cx$
Now $f (x)$ being a polynomial in $R, f (x)$ is continuous and differentiable on R. To apply Rolle's theorem.
We observe that $f (0) = 0 and f (1) = a + b + c = 0$ , by hypothesis. So there must exist at least one value of x, say $x = α \in (0, 1)$ such that
$f^{1} (α) = 0 \Leftrightarrow 3 {aα}^{2} + 2 bα + c = 0$
That is, $f^{1} (x) = 3 {ax}^{2} + 2 bx + c = 0$ has at least one root in $[0, 1]$

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