SOLUTION: Determine the value of x , where 0 ≤ x ≤ π , for which the curve y = 2cosx + 3sinx has a stationary point and determine the nature of this point.
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Question 1192776: Determine the value of x , where 0 ≤ x ≤ π , for which the curve y = 2cosx + 3sinx has a stationary point and determine the nature of this point. Answer by greenestamps(13200) (Show Source):
The stationary point is where the derivative is zero.
~ 0.983 radians
There is little value in evaluating the function by hand when x has the value arctan(3/2); use a graphing calculator to find the approximate y value at the stationary point.
The nature of the stationary point is determined by the sign of the second derivative. The graphing calculator will show that the stationary point is a maximum.
However, you can tell that the stationary point will be a maximum by looking at the second derivative of the function. The stationary point is when x is between 0 and pi/2, so we are in the first quadrant, where sin(x) and cos(x) are both positive; so the sign of the second derivative is clearly negative, which means the stationary point is a maximum.
A graph, for x from -pi/2 to 3pi/2, showing a maximum at approximately (1,3.6)....
