Given f '(x) = (x − 4)(4 − 2x), find the x-coordinate for the relative maximum on the graph of f(x).
a) 12 m/sec; 22 m/sec^2
b) 22 m/sec; 18 m/sec^2
c) 22 m/sec; 6 m/sec^2
d) 14 m/sec; 3 m/sec^2
I think you have mixed two problems.
Sorry, you're right. The question is: Given the position function s(t), s(t) = t3 − 5t, where s is measured in meters and t is in seconds, find the velocity and acceleration of the particle at t = 3 seconds.
a) 12 m/sec; 22 m/sec^2
b) 22 m/sec; 18 m/sec^2
c) 22 m/sec; 6 m/sec^2
d) 14 m/sec; 3 m/sec^2
it'd be nice if you showed some of your work. These are not hard problems.
s = t^3-5t
v = 3t^2-5
a = 6t
Now just plug in t=3
will be answer b. Sorry I just wanted to confirm. Thank you so muchhhh
To find the x-coordinate of the relative maximum on the graph of f(x), we need to find the critical points of the function and determine which one corresponds to a relative maximum.
1. First, we find the critical points by setting the derivative f'(x) equal to zero and solving for x:
f'(x) = (x - 4)(4 - 2x) = 0
2. To solve this equation, we set each factor equal to zero and solve for x:
x - 4 = 0 --> x = 4
4 - 2x = 0 --> 2x = 4 --> x = 2
3. The critical points are x = 4 and x = 2.
4. To determine which critical point corresponds to a relative maximum, we can use the second derivative test.
- If f''(x) > 0 at a critical point, it corresponds to a relative minimum.
- If f''(x) < 0 at a critical point, it corresponds to a relative maximum.
5. To find the second derivative f''(x), we differentiate the derivative f'(x):
f''(x) = d/dx [(x - 4)(4 - 2x)]
= (-2)(4 - 2x) - (x - 4)(-2)
= -8 + 4x + 2x - 8
= 6x - 16
6. Now, substitute the critical points x = 4 and x = 2 into the second derivative:
f''(4) = 6(4) - 16 = 24 - 16 = 8
f''(2) = 6(2) - 16 = 12 - 16 = -4
7. From the second derivative test, we can conclude that x = 4 corresponds to a relative minimum and x = 2 corresponds to a relative maximum.
Therefore, the x-coordinate for the relative maximum on the graph of f(x) is 2.
ANSWER: The correct option is c) 22 m/sec; 6 m/sec^2.