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Expert: Robi Bhattacharjee - 1/31/2009


Question
show that the equation X + tan(X) = 1 has at least one solution in [0,π/4]

Answer
Hi Michael,

Well, let f(x) = X + tan(X) - 1. So solving for f(x)=0 will be the same thing as solving for X in the above equation.

First note that X is continuous in the given interval. This is to say that f(x) exists for all x in [0,pi/4].

Now f(0) = -1 and f(pi/4) = pi/4. This means that by the intermediate value theorem, which states that a continuous function will reach every value between to given points, f(x) will hit zero for some x in the interval.

If you need some convincing for this theorem, try drawing a curve between the points (0,-1) and (pi/4,pi/4) without any jumps or breaks that does not go through the x axis.


I hope this helps,
Robi

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Robi Bhattacharjee

Expertise

I can answer a variety of questions on mathematics. Questions on trigonometry, calculus(preferably single variable), algebra, geometry, and number theory will be answered. I cannot answer questions on abstract branches of mathematics such as group theory. I also cannot answer questions on statistics. In number theory, I can answer questions on congruences, prime numbers, units, functions, and the riemann-zeta function.

Experience

I have studied advanced math my entire life. I started calculus in sixth grade. I have attended numerous math competitions and I am attending math organizations such as the San-Diego math circle. Also, this year I have been invited to the USAMO which is a prestigious math competition (Every year the USAMO invites 500 students from across the USA to participate in this competition. The top 6 go to represent the USA in the International Math Olympiad).

Organizations
I am in the San Diego Math Circle

Education/Credentials
I am entering high school and have received a perfect score and the STAR test 5 times in a row. I also have gotten recognitions in the AMC 10, AIME, Math Counts, and ARML. Additionally, I have won the San Diego Math Olimpiad twice in a row.

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