Math Help - differentiable function

1. differentiable function

If the function f(x) is differentiable

and

f(x) =
ax^3 - 6x ; x <= 1
bx^2 + 4 ; x > 1

then a =

A) 0
B) 1
C) -14
D) -24
E) 26

2. Originally Posted by DINOCALC09
If the function f(x) is differentiable

and

f(x) =
ax^3 - 6x ; x <= 1
bx^2 + 4 ; x > 1

then a =

A) 0
B) 1
C) -14
D) -24
E) 26
As f is differentiable f'(x)=3ax-6 for x<=1 and 2b+4 for x>1.

But both f must be continuous so at x=1 we must have:

a-6 = b+4

and as it is differentiable at x=1 we must also have:

3a-6 = 2b+4

Now solve these simultaneous equations for a.

RonL

3. is the answer C

4. Originally Posted by DINOCALC09
If the function f(x) is differentiable
and

f(x) =
ax^3 - 6x ; x <= 1
bx^2 + 4 ; x > 1
Hello,

if f is differentiable at x = 1 you have to solve for a and b:

$\begin{array}{lcr}f(1)=g(1)&\text{that means}&a-6=b+4\\f'(1)=g'(1)&\text{that means} &3a-6=2b \end{array}$

which will yield a = -14 and b = -24

So your answer is correct.

5. Originally Posted by DINOCALC09
is the answer C
Yes, the answer is C because as CaptainBlack said, to be differentiable, a function must be continuous, and therefore its derivatives should have values and be continuous. This gives you two equations with two unknowns. You probably had the f(x) values equal to each other at one, but you may have not realized that f'(x) should be continuous at that point as well.