Question

I-1 12. Determine whether The Mean Value Theorem can be applied to f(x) = 1+1 on the interval (-2, -1). If The Mean Value Theorem can be applied, find all values c that satisfy the conclusion of the theorem.

Answer 1

gies x+1 f(x)= defined on on L-2,-1 x-1 ætl Aim - Determine Mean value theorem can be applied solution - geen function f(x) = is contineoust werywhere except at se=1 But 1$ (-2,-1] f(x) is contineous on [-9,-1] X-1 80 and also fla) is differentiable in (-2,-1) There a Mean value Theorem can be applied on fra om [-9,-) then there exlist CE(-9,1) such that fico) + (-1) - f(-2) # -|- (-9) er Xti f(x) = x | da f(x) = 2+1 dX X) f'(o)= (2-1) dx 1xH) - (x+1 (x+) (x-12 fice) H-I-X-) (x-7)x1 - (x+1) [2.72 (x-7)2

SO fl(a)= (2-12 f)= (C-12 & 0 -1+1 f(-1) = f(t)=0 -3 f1-9) -2+1 = 1/3 = 1 -2-1 F1-2)= 1 3 Then from eqn 0-$ (C-1)2 1+2 2 -1/3 (C-1)2 6 = (6-1) 2 C-I = IV TCE TAV c=1tvo & (-2,-2) for positurs vale for negatiusvaly negatius cel-VÉ E(-9,-1) LV5- 2.4494 C-1 -2.4494 C = -1.4494 Argy