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The range of function $f(x ) = 4^x + 2^x + 4^{-x} + 2^{-x} + 3$ is |
$\left[\frac{3}{4},∞\right)$ $\left(\frac{3}{4},∞\right)$ $(7,∞)$ $[7,∞)$ |
$(7,∞)$ |
$4^x + 2^x + 4^{-x} + 2^{-x} + 3=t^2+\frac{1}{t^2}+t+\frac{1}{t}+3$, where $t=2^x>0, ∀\,x∈R$ $⇒\left(t+\frac{1}{t}\right)^2+\left(t+\frac{1}{t}\right)+1$ since $t+\frac{1}{t}>2, ∀\,t>0$ $∴f(x)>2^2+2+1=7$ ∴ Range is (7, ∞) |
