Question

LET
f(x) = x^4 + ax^2 + b .
The graph of f has a relative maximum at (0,1) and and inflection point at x=1 . Find the values of a and b

Answer 1

In order to do that, we should calculate the derivatives of f(x) first

Answer 2

do you remember the power rule?

Answer 3

4x^3 + 2ax ?

Answer 4

yep

Answer 5

At a maximum value, the slope of a tangent will be zero, and the derivative is the slope of the tangent, set f '(x) = 0

Answer 6

k hold on

Answer 7

\[f ' (x) = 0 = 4x^2+ 2ax\]

now calculate the second derivative

Answer 8

I set it to 0 and I am solving for what because I have a and x

Answer 9

8x + 2a

Answer 10

right,
\[f ''(x) = 8x + 2a \]

f ''(x) = 0 , when x = 1

Answer 11

the inflection point is where the second derivative = 0, the graph switches concavity

Answer 12

yeah so how do I find that , I make the second derrivative equal to 0?

Answer 13

exactly,

f ''(1) = 0

Answer 14

so I plug 1 in for a ?

Answer 15

or x

Answer 16

no, the inflection point occurs when x = 1

Answer 17

we are trying to find a

Answer 18

but I have to find the values of a and b

Answer 19

yes, we use both equations, the first derivative, and the second, 2 equations, 2 unknowns a and b

Answer 20

ok so we found the derrivatives but what is the value of a and b?

Answer 21

actually, to get b, you can just use the regular function f(x) at point (0,1)

Answer 22

f(0) = 1 = 0 + a*0 + b

b = 1

Answer 23

ok that was the normal equation but what about a ? do I use the first or second derivative

Answer 24

either one will be fine, since you are only finding one variable, i would go with the second derivative

Answer 25

look at all the ghosts

Answer 26

BOO!!!!

Answer 27

it must be North Korea

Answer 28

oh now they are going away
bye bye ghosts

Answer 29

what are ghosts ? and I do not know how to get a

Answer 30

f '' (x) = 8x + 2a

at x = 1 inflection f''(x) = 0

f ''(1) = 0 = 8(1) + 2a

Answer 31

is it 2?

Answer 32

oh

Answer 33

wait, we messed up the second derivative...the first function is to the 4th power

Answer 34

4x^3

Answer 35

f(x) = x^4 + ax^2 + b

f ' (x) = 4x^3 + 2ax

f '' (x) = 12x^2 + 2a

Answer 36

f ''(x) = 0 at x=1 inflection point

f ''(1) = 12(1)^2 + 2a = 0

12 + 2a = 0

Answer 37

a = -6

Answer 38

thank you so much!

Answer 39

you understand what they want you to know?

f ' (x) = 0 ...maximum/minimum values

f '' (x) = 0 ...inflection points

Answer 40

yes

Answer 41

but from the function in this problem, when x=0 from the point (0,1) , you can find b from just f(x) by letting x=0

Answer 42

x = 0, f(0) = b

Answer 43

some probs, you might have to use the first derivative as a maximum or minimum to find thevariable, if the function doesnt simplify like that

Answer 44

alright , I got it (: thank you !!

Answer 45

anytime, pretty fun prob there

Answer 46

i would graph it, and use a slider for a and b, you can change a and b on the fly and see what the graph does

Answer 47

https://www.desmos.com/calculator

Answer 48

https://www.desmos.com/calculator