Find the range of the function f(x) = (x^2 + 14x + 9)/(x^2 + 2) as varies over all real numbers.
+14995 Find the range of the function f(x) = (x^2 + 14x + 9)/(x^2 + 2) as varies over all real numbers.
Hello Guest!
The function f (x) = (x ^ 2 + 14x + 9) / (x ^ 2 + 2) is a hyperbolic function with two extrema.
\( f(x) = (x^2 + 14x + 9)/(x^2 + 2) \)
\(\frac{df(x)}{dx}=\frac{u'v-uv'}{v^2}\\ \frac{df(x)}{dx}=\dfrac{(2x+14)(x^2+2)-(x^2+14x+9)(2x)}{(x^2+2)^2}=0\\ 2x^3+4x+14x^2+28-(2x^3+28x^2+18x)=0\\ 2x^3+4x+14x^2+28-2x^3-28x^2-18x=0\)
\(-14x^2-14x+28=0\\ x^2+x-2=0\)
\(x=-\dfrac{1}{2}\pm \sqrt{\frac{1}{4}+2}=-0.5\pm1.5\)
\(x \in \{-2,1\}\)
\(y=(x ^ 2 + 14x + 9) / (x ^ 2 + 2) \)
\(y_{min}=-2.5\\ y_{max}=8\)
The range of the function f(x) = (x^2 + 14x + 9)/(x^2 + 2)
as varies over all real numbers is
-2.5 < R < 8
!
+267 Well, I'm late. But if other people is stuck on this problem and want a hint, here's one:
the denominator (x^2+2 in this case) can't equal 0 because then it would be undefined.
