Given the function #f(x)= 6 cos (x) #, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-pi/2, pi/2] and find the c?

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Answer 1 · 2016-12-18T17:18:33

The Mean Value Theorem has two hypotheses:

H1 : #f# is continuous on the closed interval #[a,b]#

H2 : #f# is differentiable on the open interval #(a,b)#.

#f# satisfies the hypotheses on #[a,b]# if and only if the hypotheses are true for #f# on #[a,b]# (and on #(a,b)# for H2.)

So you need to determine whether #f(x) = 6cosx# is continuous on #[-pi/2,pi/2]# and differentiable on #(-pi/2,pi/2)#.

(It is)

The conclusion on the Mean Value Theorem is

C : there is a #c# in #(a,b)# such that #f'(c) = (f(b)-f(a))/(b-a)#

To find the #c# mentioned in the conclusion, you need to solve the equation. Discard any solutions outside of #(a,b) = (-pi/2,pi/2)#.

(#c=0#)