1. Let $$ f(x)=x^{2}-8 \ln x $$. Which of the following is the root interval for $$ f(x) $$ ? $$ \begin{array}{llll} ext { a. }(0,1) & ext { b. }(4,5) & ext { c. }(2,3) & ext { d. }(3,4)\end{array} $$

Question

1. Let $$ f(x)=x^{2}-8 \ln x $$. Which of the following is the root interval for $$ f(x) $$ ? $$ \begin{array}{llll}	ext { a. }(0,1) & 	ext { b. }(4,5) & 	ext { c. }(2,3) & 	ext { d. }(3,4)\end{array} $$

UpStudy AI · Accepted Answer

We need to determine the interval in which the function

$$
f(x)=x^2-8\ln x
$$

has a root, i.e., an interval $$(a,b)$$ where $$f(a)$$ and $$f(b)$$ have opposite signs.

**Step 1. Check the Domain**

The function involves $$\ln x$$, so the domain is $$x>0$$.

**Step 2. Evaluate $$f(x)$$ at the Endpoints of Each Given Option**

The options are:

- Option a: $$(0,1)$$
- Option b: $$(4,5)$$
- Option c: $$(2,3)$$
- Option d: $$(3,4)$$

We will evaluate $$f(x)$$ at the endpoints of each interval.

**Option c: $$(2,3)$$**

- Evaluate at $$x=2$$:

$$
  f(2)=2^2-8\ln2=4-8\ln2.
  $$

Using the approximation $$\ln2\approx0.693$$:

$$
  f(2)\approx 4-8(0.693)=4-5.544\approx -1.544.
  $$

- Evaluate at $$x=3$$:

$$
  f(3)=3^2-8\ln3=9-8\ln3.
  $$

Using the approximation $$\ln3\approx1.099$$:

$$
  f(3)\approx 9-8(1.099)=9-8.792\approx 0.208.
  $$

Since $$f(2)<0$$ and $$f(3)>0$$, there is a sign change in the interval $$(2,3)$$.

**Step 3. Conclusion**

By the Intermediate Value Theorem, since $$f(x)$$ is continuous on $$(2,3)$$ and $$f(2)$$ and $$f(3)$$ have opposite signs, there must be at least one root in the interval $$(2,3)$$.

Thus, the root interval is $$(2,3)$$.
The root interval for $$ f(x) = x^2 - 8\ln x $$ is $$(2,3)$$.

1. Let \( f(x)=x^{2}-8 \ln x \). Which of the following is the root interval for \( f(x) \) ? \( \begin{array}{llll}\text { a. }(0,1) & \text { b. }(4,5) & \text { c. }(2,3) & \text { d. }(3,4)\end{array} \)

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions