Taking the derivative of a function

[ammsa]

Homework Statement

Consider the function f(x) = { ax^2 + b if x >= 1
4 - x if x<1

for which values of a and b does this function have a derivative at 1?

Homework Equations

The Attempt at a Solution

Can someone explain to me what the question wants? does it ask for the values of a and b that will make the derivative of ax^2 + b equal the derivative of 4 - x ?
I'm so lost

 

[bopll]

Homework Statement

Consider the function f(x) = { ax^2 + b if x >= 1
4 - x if x<1

for which values of a and b does this function have a derivative at 1?

Homework Equations

The Attempt at a Solution

Can someone explain to me what the question wants? does it ask for the values of a and b that will make the derivative of ax^2 + b equal the derivative of 4 - x ?
I'm so lost

you're on the right track.

This is a piecewise function, and in order for there to be a derivative at 1, the values for the derivatives AT 1 have to match from both sides...

imagine a graph of the function. you have probably learned that there will not be a derivative if there is a cusp in the graph. This is because the slope of the graph at an x value (1 for example) isn't the same from the left and the right. You want to "even out" this cusp by making the derivatives the same from both sides.

So you are definitely on the right track. You need to find the values of a and b that will make the derivative of ax^2 + b equal the derivative of 4-x. Just remember to plug in your x value. you also need to ensure the actual values connect on the graph, which is where the selection of the B value comes in (since it will disappear when you take the derivative)

 

[ammsa]

Thank you!, I just answered it, i just want to make it sure its correct.

I got the limit as x ---> 1+ of ( ax^2 + b ) = a + b
and then the limit as x ---> 1- of ( 4 - x ) = 3

to make the function continuous, a+b must = 3
------
we get the derivative of ax^2 + b, which is 2ax
and then we get the derivative of 4 -x, which is -1

at x = 1, 2ax must = -1 for the derivative to exist,
therefore 2ax = -1
a = -1/2

we plug in a in ( a + b = 3 )
-1/2 + b = 3
b = 3.5

a = -1/2
b = 3.5