If-f-x-a-e-2x-b-e-x-cx-satisfies-the-condition-f-0-1-f-log-2-31-0-log-4-f-x-cx-dx-39-2-then-

Question Number 31538 by akhilesh2684894@gmail.com last updated on 09/Mar/18

If f (x) = a e^{2 x} + b e^{x} + c x satisfies the

condition f (0) = - 1, f^{'} (\log 2) = 31,

\underset{0}{\int^{\log 4}} (f (x) - c x) d x = \frac{39}{2}, then

Answered by MJS last updated on 09/Mar/18

I hope \log = \ln, not \log_{10} \dots

f (0) = 1 \Rightarrow a + b = - 1

f^{'} (\log 2) = 31 \Rightarrow 8 a + 2 b + c = 31

\underset{0}{\int^{\log 4}} {a e}^{2 x} + {b e}^{x} d x = \frac{39}{2} \Rightarrow

\Rightarrow \frac{{a e}^{2 x}}{2} + {b e}^{x} ∣_{0}^{\log 4} = \frac{39}{2} \Rightarrow

\Rightarrow (8 a + 4 b) - (\frac{a}{2} + b) = \frac{39}{2} \Rightarrow

\Rightarrow \frac{15 a}{2} + 3 b = \frac{39}{2}

(I) a + b = - 1

(I I) 8 a + 2 b + c = 31

(I I I) \frac{15 a}{2} + 3 b = \frac{39}{2}

- 3 \times (I) + (I I I) \frac{9 a}{2} = \frac{45}{2} \Rightarrow a = 5

(I) 5 + b = - 1 \Rightarrow b = - 6

(I I) 40 - 12 + c = 31 \Rightarrow

\Rightarrow c = 3

f (x) = 5 e^{2 x} - 6 e^{x} + 3 x

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