determine the worths that a and also b which would cause ns function f(x) being differentiable in ~ x=2.

You are watching: What are the values of a and b

f(x) = { 2ax + 10 x2

Solution:

the trick below is to acquire both the attribute and it"s derivative to it is in the same in ~ ns point x = 2.

Equati~ above #1:

f1(x) = f2(x)

2ax + 10 = bx^2 - 4x + 9

bx^2 - (4x + 2ax) + (9 - 10) = 0

bx^2 - (2a + 4)x - 1 = 0

Equatitop top #2:

f1"(x) = f2"(x)

2a = 2bx - 4

a = bx - 2

us now have actually 2 equation with 2 unknowns.

bx^2 - (2a + 4)x - 1 = 0

a = bx - 2

bx^2 - (2(bx - 2) + 4)x - 1 = 0

bx^2 - (2bx - 4 + 4)x - 1 = 0

bx^2 - 2bx^2 - 1 = 0

-bx^2 - 1 = 0

bx^2 + 1 = 0

bx^2 = - 1

b = -1 / x^2

Due to the fact that x = 2,

b = -1 / 2^2

b = -1/4

a = bx - 2

a = (-1/4)(2) - 2

a = -1/2 - 2

a = -5/2

THEREFORE:

a = -5/2, b = -1/4 do f(x) differentimaybe in ~ x = 2.

f(x) = { -5x + 10 because that x 2

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the attribute demands come be consistent for ins come it is in differentiable, for this reason in ~ x=2 the equation have to equal:

2a(2)+10=b(22)-4*2+9, 4a+10=4b+1, 4a=4b-9, a=b-9/4

currently us take ns derivati have of ns attribute in ~ x=2 Because ns derivativens should be ns same:

2a=2b(2)-4, a=2b-2

currently use these 2 equation to deal with because that a and also b:

2b-2=b-9/4, b=-1/4, a=-1/4-9/4, a=-10/4

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Harold T.

the values of a and b execute no it seems ~ come fulfill both equations. A = 2b-2 for this reason a = 2(1/4) -2 = 1/2 - 4/2 = -3/2. A = -3/2 does not enhance a = -2.
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Marc L.

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