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f-x-3x-3-2x-k-has-factor-x-1-find-the-value-of-k-with-these-value-evaluate-a-d-dx-f-x-b-5-2-f-x-

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Question Number 39144 by Rio Mike last updated on 03/Jul/18
f(x) = 3x^3 − 2x + k has factor (x − 1) find the value of k. with these value evaluate a) (d/(dx ))(f(x)_ ) b) ∫_5 ^2 (f(x))
f(x)=3x3−2x+khasfactor(x−1)findthevalueofk.withthesevalueevaluatea)ddx(f(x))b)∫52(f(x))
Answered by MJS last updated on 03/Jul/18
(a) (d/dx)[f(x)]=f′(x)=9x^2 −2 (x−1)(x−α)(x−β)−(x^3 −(2/3)x+(k/3))=0 α+β+1=0 α+αβ+β+(2/3)=0 αβ+(k/3)=0 α=−β−1 β^2 +β+(1/3)=0 β^2 +β−(k/3)=0 ⇒ k=−1 f(x)=3x^3 −2x−1 (b) ∫_5 ^2 (3x^3 −2x−1)dx=[(3/4)x^4 −x^2 −x]_5 ^2 = =6−((1755)/4)=−((1731)/4)
(a)ddx[f(x)]=f′(x)=9x2−2(x−1)(x−α)(x−β)−(x3−23x+k3)=0α+β+1=0α+αβ+β+23=0αβ+k3=0α=−β−1β2+β+13=0β2+β−k3=0⇒k=−1f(x)=3x3−2x−1(b)∫25(3x3−2x−1)dx=[34x4−x2−x]52==6−17554=−17314

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