FORM GR1268 Question 17

Question:
Which of the following equations has the greatest number of real solutions?

(A) \(x^3=10-x\)

(B) \(x^2+5x-7=x+8\)

(C) \(7x+5=1-3x\)

(D) \(e^x=x\)

(E) \(\sec x=e^{-x^2}\)

Answer:
(B)

Answer Key:
Graph left-hand-side and right-hand-side functions.
(A) has one intersection.
(B) has two intersections.
(C) has one intersection.
(D) has zero intersection.
(E) has one intersection.

The only tricky one is (E).
Because \(\displaystyle\sec x = \frac{1}{\cos x}\), we have \(\displaystyle\sec(0)=\frac{1}{\cos(0)}=1\).
Because the range of \(\cos x\) is \([-1,1]\), the range of \(\sec x\) is \((-\infty, -1]\cup[1, \infty)\).

\(e^{-x^2}\) is like \(e^{-x}\), so it decays as \(x\) increases.
It also decays as \(x\) decreases because of the squaring.
It takes maximum at \(x=0\) and \(e^{-0^2}=1\).
Thus, the range of \(e^{-x^2}\) is \((0,1]\).
Since it takes the value of \(1\) only at \(x=0\), there is only one intersection with \(\sec x\).