# Math GRE subject test question

#### PsychonautQQ

Let k be the number of real solutions to the equation e^x + x - 2 = 0 in the interval [0,1] and let n be the number of real solutions NOT on the interval [0,1]. Which of the following are true?

A) k = 0 and n = 1
B) k = 1 and n = 0
C) k = n = 1
D) k > 1
E) n > 1

Can anyone help me understand this? I'm thinking that the answer is B, because if x = 1 then the equation will be greater than zero, and the equation starts below the x-axis. So somewhere in the interval [0,1], the line must cross the x=axis. Hence B is the answer. Is this correct?

#### Mark44

Mentor
Let k be the number of real solutions to the equation e^x + x - 2 = 0 in the interval [0,1] and let n be the number of real solutions NOT on the interval [0,1]. Which of the following are true?

A) k = 0 and n = 1
B) k = 1 and n = 0
C) k = n = 1
D) k > 1
E) n > 1

Can anyone help me understand this? I'm thinking that the answer is B, because if x = 1 then the equation will be greater than zero, and the equation starts below the x-axis. So somewhere in the interval [0,1], the line must cross the x=axis. Hence B is the answer. Is this correct?
Maybe or maybe not. The real question is whether the graph of the function f(x) = ex + x - 2 crosses the x-axis outside the interval [0, 1]. If it does so once, then C would be the answer.

How do you know for certain that there is only one x-intercept? Hint: take the derivative of f.

Minor quibble: an equation is not greater than zero, less than zero, or equal to zero. An equation is a statement that two quantities or expressions are equal. An inequality is a statement that one expression is larger than, or smaller than, another.

#### HallsofIvy

I think it is simpler to convert $e^x+ x- 2= 0$ to $e^x= 2- x$.
Now, even a rough graph of $y= e^x$ and $y= 2- x$ will give the answer.