Find the absolute extrema of the function on the closed interval.

minimum

Ask by Dawson Bright. in the United States

Mar 31,2025

Question

Find the absolute extrema of the function on the closed interval. $$ f(x)=\cos (\pi x), \quad\left[0, \frac{1}{6}\right] $$ minimum $$ (x, y)=(\square) $$

UpStudy AI · Accepted Answer

1. **Differentiate the function.**

We have 
   $$
   f(x)=\cos (\pi x).
   $$
   Differentiating with respect to $$ x $$ using the chain rule, we obtain
   $$
   f'(x)=-\pi\sin(\pi x).
   $$

2. **Find the critical points in the interval $$\left[0,\tfrac{1}{6}\right]$$.**

Critical points occur when $$ f'(x)=0 $$ or $$ f'(x) $$ is undefined. Since $$ f'(x) $$ is defined for all $$ x $$, we set
   $$
   -\pi\sin(\pi x)=0.
   $$
   Dividing both sides by $$-\pi$$ yields:
   $$
   \sin(\pi x)=0.
   $$
   The general solution for $$\sin(\pi x)=0$$ is
   $$
   \pi x = k\pi \quad \text{or} \quad x=k,\quad k\in\mathbb{Z}.
   $$
   Thus, $$ x=k $$. In the interval $$\left[0,\tfrac{1}{6}\right]$$, the only possible integer is $$ x=0 $$. Therefore, the only critical point in the interval is
   $$
   x=0.
   $$

3. **Evaluate $$ f(x) $$ at the critical point and endpoints.**

- At $$ x=0 $$:
     $$
     f(0)=\cos(0)=1.
     $$
     
   - At the right endpoint $$ x=\tfrac{1}{6} $$:
     $$
     f\left(\tfrac{1}{6}\right)=\cos\left(\pi\cdot\tfrac{1}{6}\right)=\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}.
     $$

4. **Determine the absolute extrema.**

Comparing the values:
   $$
   f(0)=1 \quad \text{and} \quad f\left(\tfrac{1}{6}\right)=\frac{\sqrt{3}}{2}\approx0.866.
   $$
   - The absolute maximum is at 
     $$
     (x,y)=(0,1).
     $$
   - The absolute minimum is at 
     $$
     \left(x,y\right)=\left(\frac{1}{6},\frac{\sqrt{3}}{2}\right).
     $$

5. **Answer:**

minimum  
   $$ (x, y)=\left(\frac{1}{6},\frac{\sqrt{3}}{2}\right) $$
The absolute minimum of the function $$ f(x) = \cos(\pi x) $$ on the interval $$\left[0, \frac{1}{6}\right]$$ is at $$ \left(\frac{1}{6}, \frac{\sqrt{3}}{2}\right) $$.

Find the absolute extrema of the function on the closed interval.

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Find the absolute extrema of the function on the closed interval. minimum