00:01
Okay, let's consider the question of a polynomial.
00:05
So by our condition, we know f is a polynomial of degree three.
00:13
That means we can express f as alpha times x cubed, because f is a polynomial of degree three.
00:22
So this is the highest degree.
00:24
Plus beta x squared plus gamma x plus some delta.
00:31
Okay, alpha, beta, gamma, delta are just some real numbers.
00:36
Okay, but we require alpha is not equal to zero, because otherwise f is not a polynomial of degree three.
00:45
It is a polynomial of degree two or one, which just contradicts our requirements.
00:52
So we are required to use roller's mean value theory to prove f has at most three real zeros.
01:01
Okay, now let's do it by assume not.
01:11
If not, then that means that f has more than four real zeros.
01:33
Right, let's just assume the statement is not true.
01:37
We'll get some contradiction.
01:39
Okay, let's just write down these four real zeros as a1, a2, a3, and a4.
01:51
Okay, because there are four distinct things, we can just order them so that a1 is strictly less than a2, strictly less than a3, strictly less than a4.
02:03
Okay, if the statement is not true, then we can find at least four zeros.
02:08
We just need to consider those four zeros.
02:11
By our definition, we'll have a0 zero is equal to a3 is equal to a4, because they are zero.
02:26
So the values of our function at those points are all equal to zero.
02:33
However, f is a polynomial, so we know it is differentiable and it can be differentiable.
02:43
Actually, we can say it can be differentiated by any types as we want.
02:50
So we can use the roller mean value theory.
02:54
Roller mean value theory tells us there is a b1, which is strictly greater than a1, and less than a2, and b2, which is greater than a2, and less than a3.
03:21
And so b3, which is greater than a2, and less than a3, and less than a4, such that, such that what? such that the derivative of b1, b2, b3 are all equal to zero...