How do you verify that the hypotheses of rolles theorem are right for #f(x)= x sqrt(x+2)# over the interval [2,4]?

Question

How do you verify that the hypotheses of rolles theorem are right for #f(x)= x sqrt(x+2)#   over the interval [2,4]?

Emilia Aguilar · Accepted Answer

The hypotesis of the Rolles theorem are:

If the hypotesis are satisfied, than there is a point #P(c,f(c))# such as #a < b < c# in which:

#f'(c)=0#.

Our function has domain #[-2,+oo]# (the radicand has to be positive or zero): so the function is continue in #[2,4]#.

The derivative is:

#y'=1*sqrt(x+2)+x*1/(2sqrt(x+2))#

and its domain is #(-2,+oo)#, because the root has gone at the dominator so the radicand has to be only positive: so the functionis derivable in #(2,4)#.

So the theorem can't be applicated!

Isabella Ackerman · Answer

undefined To verify that the hypotheses of Rolle's Theorem are satisfied for $ f(x) = x \sqrt{x + 2} $ over the interval $[2, 4]$, follow these steps:

1. **Check Continuity:** Verify that $ f(x) $ is continuous on the closed interval $[2, 4]$.

2. **Check Differentiability:** Confirm that $ f(x) $ is differentiable on the open interval $(2, 4)$.

3. **Check Endpoints:** Ensure that $ f(2) = f(4) $.

4. **Calculate Derivative:** Find the derivative of $ f(x) $, denoted as $ f'(x) $.

5. **Find Critical Point:** Determine if there exists a point $ c $ in the open interval $(2, 4)$ such that $ f'(c) = 0 $.

6. **Verify Hypotheses:** Confirm that all conditions are met. If all conditions are satisfied, then Rolle's Theorem is applicable.

7. **Conclusion:** If all conditions are met, conclude that there exists at least one $ c $ in the open interval $(2, 4)$ such that $ f'(c) = 0 $, implying that there exists a point $ c $ in the open interval $(2, 4)$ such that $ f'(c) = 0 $.

8. **Additional Note:** Rolle's Theorem guarantees the existence of at least one point $ c $ in the open interval $(2, 4)$ such that the derivative of $ f(x) $ is zero, given that the function meets the specified criteria.

If all steps are successfully completed, the hypotheses of Rolle's Theorem are verified for $ f(x) = x \sqrt{x + 2} $ over the interval $[2, 4]$.