Suppose f is continuous on [1,5] and the only solutions of the equation f(X)= 6 are x=1 and x=4. If f(2)=8, explain why f(3)6.

Question

Suppose f is continuous on [1,5] and the only solutions of the equation f(X)= 6 are x=1 and x=4. If f(2)=8, explain why f(3)>6.

anonymous · Answer

because since 
$$f(2)=8$$ and 
$$f(4)=6$$ and there is no other value between 2 and 4 that gives 6, f must be greater than 6 at 3. if f was lower than 6, it must have skipped over 6, which is not possible since f is continuous

anonymous · Answer

is there anyway you can draw that on a graph, so I can see it from that perspective?

jim_thompson5910 · Answer

Since f(1) = 6 and f(4) = 6, this means that f(2) is either greater than 6 or less than 6. But we know that f(2) = 8, which above 6. 


In addition, f(3) is either greater than 6 or it is less than 6. However, we know that it CANNOT be less than 6 because the jump from x=2 to x=3 would mean that the graph of f(x) would cross the line y=6 (somewhere between x=2 and x=3), but it only crosses at x=1 and x=4. 


So if f(3) was less than 6, then there would have to be a discontinuity on [1,5]. But it's stated that f is continuous on [1,5], which is a contradiction.


So f(3) > 6

anonymous · Answer

oh, ok thanks!

anonymous · Answer

It would just help me if someone could draw it on the graph

anonymous · Answer

to understand the problem more

jim_thompson5910 · Answer

take a look at the attached image for one possible graph of f(x)

anonymous · Answer

oh, ok so is this what the graph suppose to look like?

jim_thompson5910 · Answer

no, that's just one of infinitely many possible graphs of f(x)

jim_thompson5910 · Answer

the actual function does not matter as the idea of continuity is really what's important

anonymous · Answer

Ok thanks for the explanation!