2. (8 points) Using the Intermediate Value Theorem and Rolle\'s Theorem show tha

Question

2. (8 points) Using the Intermediate Value Theorem and Rolle's Theorem show that the equation sn 0 has exactly one real root orn 3. (12 points) Let f(x) = e-zsinx on (0, 2). (a) (3 pts) Find f() and f"(). Simplify. (b) (3 pts) Find the intervals of increase and of decrease of f, and all local extreme values of f ward of f, and all inflection points (x,y) of f. the points found in parts (a) and (b) (and the a-intercepts). (c) (3 pts) Find the intervals of concave upward and of concave down- (d) (3 pts) Using the above information, sketch the graph of f labeling

Explanation / Answer

2. First lets talk about the Intermediate Value Theorem.

Is days that if we have a function say F(x) and F(x) is continuous on the closed interval [a,b] and say there is a number k such that

k lies between F(a) and F(b)

Then there is a number c E [a,b] such that F(c) = k

Now we have F(x) = x/2 - sinx , x E (pi/2 , pi)

Now, F(pi/2) = pi/4 - sin(pi/2) = pi/4 - 1 < 0

and F(pi) = pi/2 - sin(pi) = pi/2 > 0

Hence when x goes from pi/2 to pi we would get atleast one such value of x at which that function F(x) cuts the x-axis.Or in other words F(x) = 0 is possible

Hence by the intermediate value theorem we could see that

there exists a number say x = k at which F(k) = 0 is possible and k lies within the interval x E [pi/2 , pi]

=> We are sure that there is atleast one real root for the given function in the given interval.

Next we need to make sure that there is exactly one root within the given interval (pi/2 , pi). For this we would use the Rolle's Theorem which says:

If F(x) is continuous within the interval [a,b] and

F(x) is differentiable within the interval (a,b) and

F(a) = F(b)

Then, there is a number c such that c lies in between a and b and F '(c) = 0.

Now, we have F(x) = x/2 - sinx

=> F '(x) = x - cosx , x E (pi/2 , pi)

Please note that the given interval is the second quadrant and in the second quadrant cosx is always negative

so the entire derivaitve F '(x) = x - cosx would always be positive in the given interval.

And as the derivative is always positive so the function F(x) is always increasing and it would never cross the x-axis again.

Hence by the Rolle's theorem there would bot be a relative maxima or relative minima as the function never crosses the x-axis.

But by the intermediate vlaue theorem we proved that the function does cross that x - axis atleast once on the interval x E (pi/2 , pi).

Hence, the function crosses the x-axis just once within the interval.

Therefore we have proved that the function has exactly one root within the given interval x E (pi/2 , pi)

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