# Find the number(s) c referred to in the intermediate valuetheorem for the function over the interval indicated and for thegiven value of L. f(x)=-x^2 +2x+3 over [-1, 0] using L=2.

Find the number(s) c referred to in the intermediate valuetheorem for the function over the interval indicated and for thegiven value of L.
$f\left(x\right)=-{x}^{2}+2x+3$ over [-1,0] using L = 2.
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Danica Ray
intermediate value theorem: Iff is continuous on a closed interval [a,b] and L is any number between f(a) andf(b) then there is at least one number c in [a,b] such that f(c) = L.
$f\left(b\right)=-\left(-1{\right)}^{2}+2\left(-1\right)+3=0$
f(a) = 0 + 0 + 3 = 3
L = 2 so it is between f(b) and f(a)
$-{x}^{2}+2x+3=2$
$-{x}^{2}+2x+1=0$
Use quadratic formula to solve: x or c = 2.4142, which is on theinterval.
(double check: $-\left(2.4142{\right)}^{2}+2\left(2.4142\right)+3=2.0000$)

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