use mean value theorem:

Question

f(x)= 7- 2^x, [0,4], c=?

Answer 1

f(4) = -9
f(0) = 6
so, we want f'(c) = -15/4

f'(x) = -ln2 2^x
-15/4 = -ln2 2^c
15/(4ln2) = 2^c
ln(15/ln16) = c ln2
c = ln(15/(4ln2))/ln2 = 2.4356

which is in the interval [0,4]