Question

Just a couple verification. PLEASE :)

Answer 1

This is about concave up and down.

\(\large\color{black}{ a) }\) when \(\large\color{black}{ f''(x)>0 }\) the \(\large\color{black}{ f(x) }\) is concave up.
\(\large\color{black}{ b) }\) when \(\large\color{black}{ f''(x)<0 }\) the \(\large\color{black}{ f(x) }\) is concave down.
\(\large\color{black}{ c) }\) when \(\large\color{black}{ f''(x)=0 }\) the \(\large\color{black}{ f(x) }\) has an inflection point e.i. a local max or min,
and that is a concavity change in a function.

Answer 2

c) is not necessarily true

Answer 3

f(x) = x^4 , f ' (x ) = 4x^3, f ' ' (x) = 12x^2 = 0 , x = 0. but 0 is not an inflection point

Answer 4

When is it not true? When the function is a parabola?

Answer 5

definition of inflection point , the concavity changes at the point.

Answer 6

yes

Answer 7

well you have to do more work to determine its a true inflection point

Answer 8

there are other cases

Answer 9

we make a sign chart and test the second derivative to the right and left of the potential inflection point

Answer 10

test values

Answer 11

but parabola is one of the cases, because it has no inflection points, since the slope is always increasing or always decreasing.

Answer 12

, no y = x^2 the slope is not always increasing

Answer 13

or y = x^4

Answer 14

yes it is

Answer 15

in y=x^2,

when the
x --> negative infinity
the slope of the function is a very big negative slope,

and then as x becomes closer to zero, the slope becomes zero,
and
as x --> positive infinity
the slope of the function is positive.

Answer 16

(sorry )