Sove for x x

Algebra Level 3

1 0 x + 1 1 x + 1 2 x = 1 3 x + 1 4 x 10^x+11^x+12^x=13^x+14^x

If A A is the number of solutions of the equation above, find ∫ 0 1 x A e d x \displaystyle\int_{0}^{1}x^{Ae}dx .


The answer is 0.269.

Sarthak Sahoo by Sarthak Sahoo , 20, India

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1 solution

Sarthak Sahoo
Dec 12, 2019

1 0 x 1 3 x + 1 1 x 1 3 x + 1 2 x 1 3 x = 1 + 1 4 x 1 3 x \dfrac{10^x}{13^x}+\dfrac{11^x}{13^x}+\dfrac{12^x}{13^x}=1+\dfrac{14^x}{13^x} On Treating the LHS and RHS separately we find that the LHS is a decreasing function and the RHS is an increasing function therefore their graphs can have at most 1 intersection point. which through inspection lies at x = 2 x=2 . Note for the problem you weren't required to find the value which satisfies the function.

1 Helpful 0 Interesting 3 Brilliant 1 Confused

I know that A is 1, but how do we get to the answer of the real question?

Saya Suka - 1 year, 6 months ago

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just integrate

Sarthak Sahoo - 1 year, 6 months ago

1 e + 1 \frac{1}{e+1} is the answer of the integrand.

Peter van der Linden - 1 year, 6 months ago

I don't remember how to, and it can't be a solution if it's not showing the final answer in it.

Saya Suka - 1 year, 6 months ago

1 pending report

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