Homework SourseUse the Intermediate Value Theorem to answer the following a
Use the Intermediate Value Theorem to answer the following:
(a) Is there any root for the equation x+sin x = x2? If so, indicate an interval
containing a root.
(b) Is there a number which is exactly 1 more than its cube? If so, indicate
an interval containing such a number.
(a) Is there any root for the equation x+sin x = x2? If so, indicate an interval
containing a root.
(b) Is there a number which is exactly 1 more than its cube? If so, indicate
an interval containing such a number.
Solution
x + sinx = x2
f(x) = x + sinx - x2
f(0) = 0 it satisfies !!!
f(1) = 1 + sin 1 - 1 = sin 1 (+ve number)
f(2) = 2 - 4 + sin2 = sin2 - 2 (-ve number)
hence there is also a root in (1,2) interval
b) x = x3 + 1
f(x) = x - x3 - 1
f(0) = -1
f(1) = -1
f(-1) = -1 + 1 -1 = -1 -ve number !!!
f(-2) = -2 + 8 - 1 = 5 +ve number !!1
using intermediatw theorem : (-2, -1) there is a root in this interval 1!!
