32 4 In this problem you will use Rolles theorem to determin

3-2 #4

In this problem you will use Rolle\'s theorem to determine whether it is possible for the function to have two or more real roots (or, equivalently, whether the graph ofcrosses the-axis two or more times). Suppose thathas at least two real roots. Choose two of these roots and call the smaller oneand the larger one. By applying Rolle\'s theorem toon the interval, there exists at least one numberin the intervalso that. The values of the derivativeare always?changingnegativezeropositiveundefined, and therefore it is?plausibleunlikelypossibleimpossibleforto have two or more real roots.

Solution

Let the two roots be (a,b).
Hence by Rolls theorem there as to be a c between a and b such that f\'(c) = 0
In this case f(x) = -6x⁵ -9x - 10
Hence f \' (x) = -30x⁴ - 9
Now as x⁴ is always positive; the values of this derivative is always negative and therefore it is impossible for f(x) to have two or more real roots