The graph of y = cos x is concave down on which interval?
(-pi,0)???????
please help!!!!
You ought to have this memorized. From zero to Pi/2 it is concave down. Or if you want to look negative, from -PI/2 to Pi/2
To determine the intervals on which the graph of y = cos(x) is concave down, we need to examine its second derivative. Let's go through the process step by step:
1. Start with the function y = cos(x).
2. Calculate the first derivative by taking the derivative of cos(x) with respect to x. The derivative of cos(x) is -sin(x). So, the first derivative is dy/dx = -sin(x).
3. Calculate the second derivative by taking the derivative of -sin(x) with respect to x. The derivative of -sin(x) is -cos(x). Therefore, the second derivative is d²y/dx² = -cos(x).
Now, we need to determine the intervals where the second derivative, -cos(x), is negative. This is done by analyzing the intervals where cos(x) is positive.
The cosine function, cos(x), is positive in the intervals [0, π/2) and (π, 3π/2], among others. In these intervals, -cos(x) will be negative.
Thus, the interval in which the graph of y = cos(x) is concave down is (π/2, π). Note that we omit points where the second derivative is equal to zero, as the question specifically asks for the interval where it is negative.
Therefore, the correct answer is (π/2, π).
I hope this explanation helps! Let me know if you have any further questions.