Question

Help me with these statements:
1)The definite integral of a polynomial function of odd degree over [-3,3] will __________ be equal to 0.

2)A function f that is continuous on [a,b] and has zeroes at a and b will __________ have a horizontal tangency at some x in (a,b).

Answer 1

These are Always, Sometimes, Never problems.

Answer 2

If c is a critical number of f and f”(c) = 2, then f _____________ has a point of inflection at x = c.

Answer 3

Think about it...

\(\large\color{#000000 }{ \displaystyle F(x)=\int_{-a}^{a} x^{2n+1}dx=\frac{1}{2n+2}x^{2n+2} }\)

And then, F(-3)=F(3), because of the symmetry of even power.
So, F(3) - F(-3) = 0


HOWEVER, some terms can be

\(\large\color{#000000 }{ \displaystyle f(x)=\int_{-a}^{a} x^{2n}dx=\frac{1}{2n+1}x^{2n+1} }\) and this is not necessarily 0

Answer 4

I am saying that sometimes it can be a poly with degree 2n+1, and another lesser term x^2n...

Answer 5

there is a test for inflection

Answer 6

let's take some sufficiently small b,
let's say

b=0.06

So, if f"(c+b) and f"(c-b) have different signs, then concavity indeed changes, and the point x=c is indeed inflection

Answer 7

Yes.

Answer 8

However, if
f"(c+b) and f"(c-b)
have same signs then x=c is NOT an inflection.

Answer 9

So your answer would be?

Answer 10

Sometimes

Answer 11

oh sorry I read
f”(c) = 0

Answer 12

if =0 , then c is possibly an inflcetion.

Answer 13

but
f”(c) = 2

never an inflcetion at x=c

Answer 14

O ok

Answer 15

I got to go...

(f is concave up at c if f''(c)=2)

Answer 16

If f(x) < 0 for all \[x \in [a,b], \] then the area between f(x) and x=0 can _____ be accurately calculated with \[\int\limits_{b}^{a} f(x) dx.\]