Which, if any, of the first four statements is false?

Question

If you think that all are true, answer with the fifth statement.

A. The graph of each polynomial, regardless of degree, must have at least one point where the tangent line is horizontal.
B. The graphs of certain polynomials can be horizontal straight lines.
C. The graphs of certain polynomials can be slanted (but not vertical) straight lines.
D. If a polynomial is of degree n , then its graph can have, at most, n - 1 points with a horizontal tangent line.
E. All of the foregoing are true.

Answer 1

B, C, and D are true.
Counter example for A would be any straight (non vertical) line.
y = 2x + 3 does not have a horizontal tangent.
y' = 2 which never equals zero.
An example for B would be
y = 5 (or any other real number)
C) y = 2x +3 is a slanted straight line.
D) This follows from the statement that a polynomial of degree n has at most (n-1) turning points.

Answer 2

A. false
For instance, y = ax has no horizontal tangent for any a<>0

B. true
If you accept polynomials of degree zero.

C. true
All polynomials of degree one

D. true
The horizontal tangent of f(x) is found by solving f'(x) = 0, a polynomial equation of degree n-1. Such an equation has precisely n-1 solutions (not necessarily distinct) over a complex number field, and therefore at most n-1 real solutions.

Answer 3

A. This is false since linear functions are polynomials with no horizontal tangent line everywhere.
B. This is true. The constant function is a polynomial similar with the linear but with zero slope.
C. This is true. Linear functions is one of the examples.
D. This is true. On their extrema, whether maximum or minimum, we can find a horizontal tangent line.

Answer 4

I'd say that D is the best answer for a true statement. Last I knew straight lines were mononomials.

Answer 5

false: A (eg y=x) E
true: B C D

Answer 6

A is false.