SOLUTION: Find a and b so that the function f(x)={3x^3−8x^2+8;x<−2 ------{ax+b ;x≥−2 is both continuous and differentiable. a= b=

Algebra ->  Expressions-with-variables -> SOLUTION: Find a and b so that the function f(x)={3x^3−8x^2+8;x<−2 ------{ax+b ;x≥−2 is both continuous and differentiable. a= b=      Log On


   

Question 1183693: Find a and b so that the function
f(x)={3x^3−8x^2+8;x<−2
------{ax+b ;x≥−2
is both continuous and differentiable.
a=
b=
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!


The polynomial 3x%5E3-8x%5E2%2B8 takes on the value 
3%28-2%29%5E3%E2%88%928%28-2%29%5E2%2B8
3%28-8%29-8%284%29%2B8
-24-32%2B8
-48 when x=-2

The polynomial 3x%5E3-8x%5E2%2B8 also has derivative
9x%5E2-16x and when x=-2, the derivative is
9%28-2%29%5E2-16%28-2%29
9%284%29%2B32
68

So the line y+=+ax%2Bb must join the polynomial y+=+3x%5E3-8x%5E2%2B8 
at the point (-2,-48) in order for f(x) to be continuous there.  
Also the line must have the same slope as the polynomial has 
derivative there.  

y = ax + b  has slope a, so we have a=68

y = 68x + b

and it must pass through (-2,-48), so we substitute:

-48 = 68(-2) + b
-48 = -136 + b
88 = b

So a = 68 and b = 88.

Edwin