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Please help with this Intermediate value theorem!?
The question is:
Verify that the intermediate value theorem applies and show that somewhere in the interval [1,2] there is a root of h(x) = -15x^3 +6e^x. Note that you are not being asked to find the root, only to show that one exists.
Please help answer and explain if possible. Thanks
1 Answer
- Meng tianLv 78 years agoFavorite Answer
h(x) is the sum of a polynomial function (-15x^3) and an exponential function (6e^x).
Polynomial functions and exponentials functions are continuous everywhere, so the resulting function h(x) is continuous everywhere as well, in particular on [1, 2].
h(1) = -15 + 6e^1 > 0
h(2) = -120 + 6e^2 < 0
By IVT, h(c) = 0 for some c, where 1 <= c <= 2, i.e. there is a root in [1, 2].