Question

how do i start this problem? if f(x)=x^2+sinx, show that there must be a number c such that f(c)=50
help please

Answer 1

is it f(c) = 50

OR

is it f ' (c) = 50 ... notice the prime notation

??

Answer 2

f(c)=50

Answer 3

so it sounds like you need to solve

f(x)=x^2+sinx

f(c)=c^2+sin(c)

50 = c^2 + sin(c)

0 = c^2 + sin(c) - 50

you can't do this analytically, but you can get an approximation with a graphing calculator

Answer 4

you don't need to actually solve for c though

you just need to show that such a value of c exists to make 0 = c^2 + sin(c) - 50 true

Answer 5

so that means you can use the intermediate value theorem

do you know what I'm referring to?

Answer 6

don't you need interval values to use ivt?

Answer 7

yeah so you'll have to use a graphing calculator to spot the left and right boundaries of the interval

Answer 8

i see im going to use geogebra be right back

Answer 9

Use this if you don't have a graphing calculator

https://www.desmos.com/calculator

Answer 10

geogebra works too (actually works better)

Answer 11

use x instead of c when it comes to graphing

so graph x^2+sin(x)-50

Answer 12

ok

Answer 13

it must be from -7 to 7 i think

Answer 14

you should see 2 x-intercepts

Answer 15

what's a good interval for the x-intercept on the left

Answer 16

yeah one at -7 and the other at 7

Answer 17

on the left at -7

Answer 18

it's close to -7, but not at -7

Answer 19

the left x-intercept is between -8 and -7

Answer 20

i see

Answer 21

so if g(x) = x^2+sin(x)-50

then calculate g(-8) and g(-7)

you should see a sign change

Answer 22

the sign change will indicate there is at least one root for g(x) somewhere between -8 and -7

Answer 23

ok

Answer 24

the other root is pretty close to +7, but not quite fully at +7

it's between +7 and +8

calculate g(7) and g(8) and you should see a sign change

Answer 25

g(-8) is about 13.01 and g(-7) -1.66

Answer 26

good and good

Answer 27

the sign change from + to - indicates we have at least one root in between there

Answer 28

assuming g(x) is continuous (which it is)

Answer 29

g(7) is -0.34 and g(8) is 14.99

Answer 30

very good

Answer 31

so this shows that there are at least 2 values of c that make f(c)=50 true

we have the graph to definitively say exactly 2, but you would use the mean value theorem to prove that (which would not involve the graph at all)

Answer 32

ok