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
Meng tian
Favorite 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].