Thread: Calculus: True or false statements

1. Calculus: True or false statements

Let p(x) be a polynomial function of degree n. Determine if each of the following statements is TRUE or FALSE and justify your answer in writing.
NOTE: If you can find a counterexample, the counterexample is enough to justify saying the statement is FALSE.

(a) p(x) has a y intercept.

(b) If n is odd then p(x) has at least one x-intercept

(c) If n is 2, then p (x) is always concave up.

(d) p(x) is continuous at all real numbers (i.e. on the interval (negative infinity, positive infinity)

My teacher has not taught us anything so I am completely lost. However I did try to figure and this is how far I got:

(a) y= x^2+2
y=2

(b) Im not sure of this one but i think there is at leats one x intecerpt

(c) He didn't teach us concave. Other than it is like water. Meaning it is either low or high and overflowing

(d)Wouldn't this depend on where the equation starts? Like if it starts in positive infinity it was only be positive infinity

2. Originally Posted by asweet1
Let p(x) be a polynomial function of degree n. Determine if each of the following statements is TRUE or FALSE and justify your answer in writing.
NOTE: If you can find a counterexample, the counterexample is enough to justify saying the statement is FALSE.

(a) p(x) has a y intercept.

(b) If n is odd then p(x) has at least one x-intercept

(c) If n is 2, then p (x) is always concave up.

(d) p(x) is continuous at all real numbers (i.e. on the interval (negative infinity, positive infinity)

My teacher has not taught us anything so I am completely lost. However I did try to figure and this is how far I got:

(a) y= x^2+2
y=2
A single example does not prove the proposition. This is true because the the y-intercept is the value of the polynomial when x=0. Polynomials are defined for all values of their argument so y=P(0) is well defined and is the y-intercept.

(b) Im not sure of this one but i think there is at leats one x intecerpt
Is true. If the degree is odd then the polynomial has different signs at P(x) and P(-x) for x large enough, but a polynomial is continuous and hence if its sign changes over (-x,x) it has a root in the interval.

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