59. (a) Prove that the equation has at least one real root. (b) Use your graphing device to find the

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59.

(a) Prove that the equation has at least one real root.
(b) Use your graphing device to find the root correct to three decimal places.

100e^(-x/100)=0.01x^2

Calculus: Early Transcendentals
Chapter 2: Limits and Derivatives
Section 2.5: Continuity
Problem 59

7/5/2024 - 3,453 Subscribers - 780,960 Views
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If the question were simply finding the real roots, then after using square root u get 2 cases:
I. 100 = xe^(x/200) where x>0 and II. 100 = -xe^(x/200) where x < 0
x =0 obviously doesn't count

Both of them can be solved by using Lambert's function W(xe^x) = x:
I. x = 200W(0.5)
II. x = 200W(-0.5)

The second equation gives us 2 roots <0 since -0.5 is bigger than -1/e and less than 0

bratdarishki