For theta frac{pi}{3},the value of f(theta) | vedclass

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(D) Given $f(	heta) = \sec^2 	heta + \cos^2 	heta$.We know that for any real $	heta$,$\sec^2 	heta \ge 1$ and $0 < \cos^2 	heta \le 1$.Specifica

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$[2, \infty)$

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(D) Given $f(	heta) = \sec^2 	heta + \cos^2 	heta$.We know that for any real $	heta$,$\sec^2 	heta \ge 1$ and $0 Specifically,for $	heta > \frac{\pi}{3}$,we have $\sec 	heta > \sec(\frac{\pi}{3}) = 2$,so $\sec^2 	heta > 4$.However,using the $AM$-$GM$ inequality for positive terms $a = \sec^2 	heta$ and $b = \cos^2 	heta$:$f(	heta) = \sec^2 	heta + \cos^2 	heta \ge 2 \sqrt{\sec^2 	heta \cdot \cos^2 	heta} = 2 \sqrt{1} = 2$.The minimum value is $2$ when $\sec^2 	heta = \cos^2 	heta$,which implies $\cos^4 	heta = 1$,or $\cos^2 	heta = 1$ (i.e.,$	heta = 0$,but here $	heta > \frac{\pi}{3}$).As $	heta 	o \frac{\pi}{2}$,$\sec^2 	heta 	o \infty$,so $f(	heta) 	o \infty$.Thus,the range is $[2, \infty)$.

For $\theta > \frac{\pi}{3}$,the value of $f(\theta) = \sec^2 \theta + \cos^2 \theta$ always lies in the interval

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