Posted by **Yoona** on Monday, November 21, 2011 at 8:25am.

let f be the function defined by

|x-1|+2 for X<1

f(x)=

ax^2-Bx, for X>or equal to 1. where a and b are constants

a)if a=2 and b=3 is f continious for all x? justify your answer

b)describe all the values of a and b for which f is a continious function

c) For what values of a and b is f both continious and differentiable?

- AP Calculous -
**Yoona**, Monday, November 21, 2011 at 8:26am
f(x) is both functions.. I don't know why the system didn't let me keep the spaces to show that

- AP Calculous -
**Steve**, Monday, November 21, 2011 at 10:44am
First off, stop misspelling "calculus".

Now, if a=2 and b=3, we have

f(x) = 2x^2 - 3x

f(1) = -1

But lim x->1- = |1-1| + 2 = 2

So, f is not continuous at x=1.

__________________

If f is to be continuous, it needs to be continuous at x=1. It is continuous everywhere else already.

So, ax^2 - bx must = 2 at x=1

a - b = 2

So, there are any number of parabolas which will make f continuous at x=1.

6x^2 - 4x

-3x^2 + 5x

etc.

__________________________

Now, for f to be also differentiable, the slopes must match at x=1

The slope of |x-1| is -1 when approaching from the left.

ax^2 - (a-2)x must also have slope -1 at x=1

f'(x) = 2ax - (a-2)

f'(1) = 2a - a + 2 = -1

a = -3

So,

f(x) = |x-1| + 2 for x<1

f(x) = -3x^2 + 5x for x>=1

is both continuous and differentiable everywhere.

## Answer This Question

## Related Questions

- Calculous - let f be the function defined by |x-1|+2 for X<1 f(x)= ax^2-Bx, ...
- Calculus - Functions? - #1. A cubic polynomial function f is defined by f(x) = ...
- Calculous - A dunction is continious on the closed interval [-3,3] such that of ...
- Calculous - the figure shows the graph of F', the derivative of a function f. ...
- Calculus c - Let f be a twice-differentiable function defined on the interval -1...
- calc - Let f be the function defined by f(x) = (x 2 + 1)e -x for -4 < x < ...
- Calculus - Let f be a twice-differentiable function defined on the interval -1.2...
- Calc 12 - Let f be the function defined by f(x) = (x^2 + 1)e^-x for -4≤x&#...
- Calculus - Let g and h be any two twice-differentiable functions that are ...
- calculus - Let f be the function defined by f(x)= sqrt(x), 0 <or= x <or= 4...

More Related Questions