1.Let f(x)={6x?3 if x?9 ,?6x+b ,if x>9 If f(x) is a function which is continuous

Question

1.Let f(x)={6x?3 if x?9

,?6x+b ,if x>9
If f(x) is a function which is continuous everywhere, then we must have
b=

2. Let f be a continuous function such that f(5)<0<f(7). Then the Intermediate Value Theorem implies that f(x)=0 on the interval (A,B) where A=  and B=

3.A function f(x) is said to have a jump discontinuity at x=a if:
1. limx?a?f(x) exists.
2. limx?a+f(x) exists.
3. The left and right limits are not equal.
Let f(x)=???x2+3x+3,9,?7x+6,if x<2if x=2if x>2
Show that f(x) has a jump discontinuity at x=2 by calculating the limits from the left and right at x=2.
limx?2?f(x)=
limx?2+f(x)=
Now for fun, try to graph f(x).

Explanation / Answer

for f(x) to be continuous it should be continuous at x=9

51 = -6*9 + b

b = 105

at x=5, f(x) < 0 and x=7, f(x)>0

therefore the curve crosses the x axis somewhere between 5 and 7,

A=5, B=7

limx?2?f(x)= (2)^2 + 3*2 + 3 = 13

limx?2+f(x)= -7(2) + 6 = -8

and at x=2, f(x) = 9

hence limx?2?f(x) not equal to limx?2+f(x)

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