Calculus true or false?

Question

TRUE OR FALSE if false explain why
1. if graph of function has 3 x-intercepts, yhen it must have atleast 2 points at which its tangent line is horzontal.

2. If graph of a polynomial function has 3 x-intercepts then it must have atleast 2 points at which tangent line is horizontal

3. If a function is continuous on closed interval, then it must have a minimum on the interval.

4. every nth-degree polynomial has (n-1) crtical numbers

5.An nth-degree polynomial has at most (n-1) critical numbers

Answer 1

1. False. Think of a graph made up of straight
line segments which crosses the x-axis 3 times.
2. True. A polynomial function is continuous and
differentiable everywhere, so Rolle's theorem
gives this result.
3. True. This is part of the extreme value theorem.
The function also has a maximum on the interval
4. False. A critical point of a polynomial is a place
where the derivative is 0. The polynomial
x³+x has no critical points.
5. True. The derivative of an nth degree polynomial
is a polynomial of degree n-1, so it has
at most n-1 real roots.

Answer 2

3. If a function is continuous on closed interval, then it must have a minimum on the interval.

True. Either it has a critical point, or one of the endpoints is the minimum. This is not necessarily true for an open interval.

4. every nth-degree polynomial has (n-1) crtical numbers

Consider n^4. This is a degree 4 polynomial, but only has one critcal value, at x=0.

5.An nth-degree polynomial has at most (n-1) critical numbers

This is true, because its derivative is an (n-1)th degree polynomial which can have at most n-1 roots, and every critical value must be a root of the derivative.

Answer 3

1. False

If the function is discontinuous it could easily pass through the x-axis the times without ever creating a derivative (tangent line).

2. True.

Polynomials are always continuous.

If the function is switching from positive to negative (which it is because it passes through the x-axis three times), then it is increasing and decreasing along portions of the graph. When a function switches from increasing to decreasing the site of the change is called a critical points. Critical points can be maxima, minima, or inflection points. At each of these points the derivative of the graph (tangent line) is equal to zero. (this means it is horizontal.)


Sorry I can't help you with the others

Answer 4

true