1. Important Values for Continuous functions

2. If a function assumes values f(a)=10, and f(b)=15, does the function need to equal 12 at some point?
What if the function is continuous?

3. Determine if f(x) equals k on the given interval
f(x)= 2𝑥 𝑥 ; k=8, [1,3]

4. Determine if f(x) equals k on the given interval
f(x)= 12𝑥 7 −200 37𝑥 ; k=1, [1,3]

5. Intermediate Value Theorem
If f is a continuous function on [a,b], and takes on values f(a) and f(b), then it also assumes all values between f(a) and f(b) on that interval
Corollary: If a<b, and f(a) and f(b) have opposite signs, then f has at least one zero between a and b.

6. Determine if this equation has any roots on [-1, 2]
𝑥 8 = 2 𝑥

7. Determine if this equation has any roots on [–π, π]
𝑠𝑖𝑛𝑥=𝑥+2

8. Determine if f(x) equals k on the given interval
f(x)= 1 𝑥−3 +2; k=2, [0,5]

9. Determine if this equation has any roots on [0,3]
𝑓 𝑥 = 𝑥+1 3 −4

10. Determine if this equation has any roots on [0,3]
𝑓 𝑥 = 𝑥 2 +3𝑥+2

11. Determine if this function will equal -10 on [0, 5]
𝑓 𝑥 =20 sin 𝑥+3 cos⁡( 𝑥 2 2 )

13. Determine if there are any zeros for each function

14. Extreme Value Theorem
If f is a continuous function whose domain is a closed interval, [a,b], then f has a an absolute maximum and minimum on [a,b]

15. Find the extreme values of the function for [5, 6]
𝑓 𝑥 =− 𝑥 2 +11𝑥−30

16. Find the extreme values of the function for [0, 2]
𝑓 𝑥 =− 𝑥 2 +11𝑥−30

17. Find the extreme values of the function for [0,5]
𝑓 𝑥 = 𝑥+1 3 −4