1) The Mean Value Theorem says there must be a number c in the interval (0,2pi)
ID: 3343126 • Letter: 1
Question
1) The Mean Value Theorem says there must be a number c in the interval (0,2pi) at which the tangent line to f(x) = cosx - sinx is parallel to the secant line for the graph joining the points (0, f(0)) and (2pi, f(2pi)). In fact, for this example there are two such numbers c. Find them both.
2) Find the critical points for f(x) = x + cos x, 0 %u2264 x %u2264 2%u03C0, Determine the intervals where f is increasing or decreasing, Classify each critical point as local maximum, local minimum, or neither one, Determine the intervals where f is concave up and where it is concave down, Determine any points of in%uFB02ection for f.
Please answer neatly...Thank You!
Explanation / Answer
please follow this....
f(x) = cos(x) - sin(x)
f'(x) = -sin(x) - cos(x)
First, find the slope of the secant line:
(f(2pi) - f(0)) / (2pi - 0) =>
(cos(2pi) - sin(2pi) - cos(0) + sin(0)) / (2pi) =>
(1 - 0 - 1 + 0) / 2pi =>
0
f'(x) = 0
0 = -sin(x) - cos(x)
sin(x) = -cos(x)
tan(x) = -1
x = 3pi/4 , 7pi/4
c = 3pi/4 and 7pi/4
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