Questions: Graf ktorej z nasledujúcich funkcií má na intervale ⟨0,2 π⟩ najviac priesečníkov s osou x ? (A) f1: y=2+sin x (B) f2: y=2 sin x (C) f3: y=sin (x/2) (D) f4: y=sin 2x (E) f5: y=sin x

Solution Steps

To determine which function has the fewest intersections with the x-axis on the interval $\langle 0, 2\pi \rangle$, we need to find the number of roots (or zeros) of each function within this interval. A function intersects the x-axis where its value is zero. We will evaluate each function to find the number of such points.

We are given five functions to analyze for their intersections with the x-axis on the interval $\langle 0, 2\pi \rangle$:

• $f_1: y = 2 + \sin x$
• $f_2: y = 2 \sin x$
• $f_3: y = \sin \frac{x}{2}$
• $f_4: y = \sin 2x$
• $f_5: y = \sin x$

We evaluate each function to determine the number of roots (intersections with the x-axis) within the specified interval:

• For $f_1$, $2 + \sin x = 0$ has 0 intersections.
• For $f_2$, $2 \sin x = 0$ has 2 intersections.
• For $f_3$, $\sin \frac{x}{2} = 0$ has 2 intersections.
• For $f_4$, $\sin 2x = 0$ has 2 intersections.
• For $f_5$, $\sin x = 0$ has 2 intersections.

From the counts:

• $f_1$: 0 intersections
• $f_2$: 2 intersections
• $f_3$: 2 intersections
• $f_4$: 2 intersections
• $f_5$: 2 intersections

The function $f_1$ has the fewest intersections with the x-axis.

Final Answer