Question:

Step 1: Rewrite the function using trigonometric identities. We start with $f(x) = \dfrac{1}{\cos^2 x} + \dfrac{4}{\sin^2 x}$, which can be rewritten using the definitions of secant and cosecant: $$f(x) = \sec^2 x + 4\csc^2 x$$ Now, apply the Pythagorean identities $\sec^2 x = 1 + \tan^2 x$ and $\csc^2 x = 1 + \cot^2 x$: $$f(x) = (1 + \tan^2 x) + 4(1 + \cot^2 x)$$ $$f(x) = 1 + \tan^2 x + 4 + 4\cot^2 x$$ $$f(x) = 5 + \tan^2 x + 4\cot^2 x$$ Step 2: Apply the AM-GM inequality to minimize the variable part. To find the minimum value of $f(x)$, we need to minimize $\tan^2 x + 4\cot^2 x$. Using the AM-GM inequality: $$\tan^2 x + 4\cot^2 x \geq 2\sqrt{\tan^2 x \cdot 4\cot^2 x}$$ $$\tan^2 x + 4\cot^2 x \geq 2\sqrt{4 \cdot \tan^2 x \cdot \cot^2 x}$$ Since $\tan x \cdot \cot x = 1$, we have $\tan^2 x \cdot \cot^2 x = 1$: $$\tan^2 x + 4\cot^2 x \geq 2\sqrt{4 \cdot 1} = 2\sqrt{4} = 4$$ Step 3: Determine when equality holds in the AM-GM inequality. Equality in AM-GM holds when the two terms are equal: $$\tan^2 x = 4\cot^2 x$$ Multiplying both sides by $\tan^2 x$: $$\tan^4 x = 4$$ $$\tan^2 x = 2$$ This occurs at a valid value of $x$ in the interval $\left(0, \dfrac{\pi}{2}\right)$. Step 4: Calculate the minimum value of $f(x)$. Substituting the minimum value of $\tan^2 x + 4\cot^2 x = 4$ into our expression: $$f(x)_{\min} = 5 + 4 = 9$$ **Final Answer:** The minimum value of $f(x)$ is $\boxed{9}$, which corresponds to **Option 3**.