Question

Suppose f(x) is continuous on [2,8] and −4≤f′(x)≤5 for all x in (2,8). Use the Mean Value Theorem to estimate f(8)−f(2).

Answer 1

could be between -24 to 60

Answer 2

-24 works

Answer 3

??

Answer 4

-24 to 30

Answer 5

experimentx's typo :)

Answer 6

any idea chlorophyll??

Answer 7

thanks chlorophyll

Answer 8

one last question

By applying Rolle's theorem, check whether it is possible that the function f(x)=x^9+x−18 has two real roots.

Answer: (input possible or impossible )...ans is impossible

Your reason is that if f(x) has two real roots then by Rolle's theorem:
f′(x) must be (input a number here)

Answer 9

-4 = < Δy/ Δx <= 5
-4 * 6 = < Δy < = 5 * 6

Answer 10

i need help wit the second part pls

Answer 11

you sure the answer is that??

Answer 12

-24 t0 30 worked

Answer 13

\[2 \le x \le 8\]
\[-4 \le f'(x) \le 5\]
f is continuous in differentable on [2,8] so we know there is c btw 2 and 8 such that
\[f'(c)=\frac{f(8)-f(2)}{8-2}\]
And we know
\[-4 \le f'(c) \le 5 \]
=> we have
\[-4 \le \frac{f(8)-f(2)}{8-2} \le 5 \]

Answer 14

\[-4 (8-2) \le f(8)-f(2) \le 5(8-2)\]

Answer 15

and not in*
sorry type-o there
my english sucks but the math is good lol

Answer 16

lol thanks.

can you help me with this last one

Answer 17

By applying Rolle's theorem, check whether it is possible that the function f(x)=x^9+x−18 has two real roots.

Answer: (input possible or impossible )...ans is impossible

Your reason is that if f(x) has two real roots then by Rolle's theorem:
f′(x) must be (input a number here

Answer 18

was it a parabola?? i thought i saw ^9 up there

Answer 19

well, that was interestig,
f(c) would give the value of vertex. if negative, and the parabolas is concave upwards then two real roots.

Answer 20

and if f(c) positive and the paraboal concave downwards then also two real roots. if f(c) is zero than single roots ... but i was wondering where would we apply Rolle's theorem.

Answer 21

i guess it would be just to assume that f(a) = f(b) = 0 , and a,b be the roots real roots
i.e. point c implies point a and point b on x-axis??
but what to do if there're were single root??

Answer 22

zero works thanks very much for the help

Answer 23

point a, b, and c points coincide ... well that atleast work for parabola.

Answer 24

the real problem lies in this type of graph

that was a parabola, we knew that if it crosses x-axis then it means either it opens upward or downward ... but i really don't know what to do about this type of graph.