Question

Sketch a graph of one function, f(x), that satisfies all of the

following:

Answer 1

The function passes through (0, 4) and (6, 0) and f'(x) < 0 for x < 2 and x > 4 shows that the function is decreasing on the intervals (-∞, 2) and (4, ∞).

Intercepts: The function passes through (0, 4) and (6, 0) due to the conditions f(0) = 4 and f(6) = 0.

Intervals of increasing/decreasing:

f'(x) < 0 for x < 2 and x > 4 shows that the function is decreasing on the intervals (-∞, 2) and (4, ∞).

f'(x) > 0 for 2 < x < 4 shows that the function is increasing on the interval (2, 4).

Critical points:

f'(2) does not exist indicates a sharp turn or a vertical asymptote at x = 2. The graph shows a sharp turn (cusp) at this point.

f'(4) = 0 indicates a potential extremum (maximum or minimum) at x = 4. Since we transition from increasing to decreasing at x = 4, it becomes a maximum point.

Smoothness: The function is expected to be smooth everywhere except at x = 2, where it has a sharp turn.

Answer 2

u can draw it everything is given