The questionThe involves uploaded identifying image the contains local a extrema problem of regarding the finding function local $extrema f ((xmax)ima = and \ minimasqrt){ for5 a} tr \igoncosometric( function3 andx selecting) the + appropriate \ graphsin of( the3 functionx and) its$ on the interval $[0, 2\pi] derivative$, analyzing its derivative,.

andHere is a selecting detailed the corresponding graph.

breakdown### Steps of to the Solve:

solution1:

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**###Determine Critical Points Step **1: : - Understanding Compute the the derivative Function ( fThe'( givenx function) is : ) of ( f(x[ ) =f \sqrt(x{)5} \cos(3x =) \ + \sin(sqrt3{5x}) \ \cos). (3x) + \sin( -3 Setx) ] (The task involves: 1 f. Finding the '(local maxima and minimax ()if = any) and0 their ) to corresponding ( findx\ critical points)-values where the. slope2 of. the Selecting tangent the is graph that zero. correctly represents - the Solve function for \(f x(x)) in and ( its [0, 2\ derivativepi (]f '().

x2).).

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Class###ify Step Ext rema2**:: Critical Points and Extrema To find local extrema:

1. ** -Compute Use the the second derivative derivative ( testf or'( analyzex ()\ f):'(** x ) \ [ ) sign f changes'( aroundx) the = critical \ pointsfrac to classify them{ as locald}{ maxima ordx} minima.

2. [ \sqrtEvaluate { f(x)5 at} Critical \ Pointscos(: 3x) + \sin( -3 Calculatex ()] f (x ) = \ -3)\ atsqrt the{ critical5 points} and \ determinesin their( values3.

x4). + ** 3Graph\cos(3 Matchingx) **: \ -]

Match2 the. behavior ** ofSet \( ff(x'()x) = and $0 f\'()x to) find$ critical to the given graphs (A, B, points: C** , $$-3\sqrt{5} \ D).

sin(3x) +Would 3\cos( you3x) = like0 me Simplify to: solve this \ step[ by step \ intan detail(3 orx directly) proceed = to \frac{\sqrt{ the5}} derivative{ calculation1?} \] Solve forx$within the interval$0 \leq x \leq \frac{\pi}{2}$:$3x = \arctan(\sqrt{5}) \implies x = \frac{\arctan(\sqrt{5})}{3}$$ (and possibly additional points based on periodicity).

3. Second derivative test (or analyze $f'(x)$'s behavior around critical points) to classify critical points as maxima or minima.

Step 3: Select the Correct Graph

The options in the image show graphs of $f(x)$ and $f'(x)$. Based on the derivative $f'(x)$, key features to observe include:

1. Where $f'(x) = 0$ (critical points),
2. Where $f'(x) > 0$ (function is increasing),
3. Where $f'(x) < 0$ (function is decreasing).

Analyzing these features will help match the correct graph.

Would you like me to:

• Fully calculate the $x$-values and classify extrema?
• Match the correct graph choice for $f(x)$ and $f'(x)$?
• Provide step-by-step visuals or additional clarifications?

Related Questions

1. What is the relationship between $f'(x)$ and the shape of $f(x)$?
2. How do periodic functions like $\sin(x)$ and $\cos(x)$ influence critical points?
3. What additional conditions could cause $f(x)$ to have no local extrema?
4. Why does the interval $0 \leq x \leq \pi/2$ matter in this problem?
5. How does the amplitude of $\sqrt{5}$ affect the function's extrema?

Tip

Always analyze the periodicity of trigonometric functions when solving for critical points, as they often repeat within the given interval!