If you can answer this calc question, you are officially a genius?

Question

If f(x) is continuous and differentiable and

f(x) = ax^4 + 5x , when x is less than or equal to 2
f(x) = bx^2 - 3x , when x is greater than 2

(note: the function is a piecewise function)
then b equals what?

Show work, please and thank you soooooo much.

Answer 1

a = .5
b = 6

I basically used system of equations. because the piecewise function must be continuous and differentiable, the two functions must have the same value when x is 2 and must have the same derivative with x is 2.

Answer 2

Take the 1'st derivative of each function and set them equal to each other. Solve for b in terms of a when x = 2. Done ☺


Doug

Answer 3

f(x) = ax^4 + 5x, x ≤ 2
f(x) = bx² - 3x, x > 2

f(2) = 16a + 10
When x > 2
lim x→2 f(x) = 4b - 6 = 16a + 10 for continuity

ie 16a - 4b = -16
ie 4a - b = -4 ........ (1)

f'(x) = 4ax³ + 5, x ≤ 2
f'(x) = 2bx - 3, x > 2

f'(2) = 32a + 5
When x >2
lim x→2 f'(x) = 4b - 3 = 32a + 5 for differentiability

ie 32a - 4b = -8
ie 8a - b = -2 .......... (2)

Equation (2) - equation (1)
4a = 2
So a = ½
thus b = 6

and so
f(x) = ½x^4 + 5x, x ≤ 2
= 6x² - 3x, x > 2

Answer 4

do they give you what a equals to

if they do, just set the equations equal to each other and plug in 2 for x

Answer 5

If you did your own homework, maybe YOU could be the genius here.

Answer 6

you must be indian LOL