# How do you find the intervals of increasing and decreasing using the first derivative given #y=sin^2x+sinx# in #0 ≤ x ≤ (5pi)/2#?

Distinguish:

The function will therefore shift directions at each of these points.

I hope this is useful!

By signing up, you agree to our Terms of Service and Privacy Policy

To find the intervals of increasing and decreasing for the function ( y = \sin^2(x) + \sin(x) ) on the interval ( 0 \leq x \leq \frac{5\pi}{2} ), you first need to find the first derivative of the function. Then, determine where the first derivative is positive (indicating increasing) and where it is negative (indicating decreasing).

The first derivative of ( y = \sin^2(x) + \sin(x) ) with respect to ( x ) is:

[ y' = (2\sin(x)\cos(x)) + \cos(x) ]

Now, set ( y' ) equal to zero and solve for ( x ) to find critical points. After that, determine the sign of ( y' ) in the intervals between these critical points and at the endpoints of the given interval to identify where the function is increasing or decreasing.

The critical points can be found by solving ( y' = 0 ):

[ 2\sin(x)\cos(x) + \cos(x) = 0 ] [ \cos(x)(2\sin(x) + 1) = 0 ]

This equation equals zero when ( \cos(x) = 0 ) or ( 2\sin(x) + 1 = 0 ).

For ( \cos(x) = 0 ), the solutions are ( x = \frac{\pi}{2} ) and ( x = \frac{3\pi}{2} ).

For ( 2\sin(x) + 1 = 0 ), the solution is ( x = \frac{7\pi}{6} ).

Now, test the intervals ( 0 \leq x \leq \frac{\pi}{2} ), ( \frac{\pi}{2} \leq x \leq \frac{7\pi}{6} ), ( \frac{7\pi}{6} \leq x \leq \frac{3\pi}{2} ), and ( \frac{3\pi}{2} \leq x \leq \frac{5\pi}{2} ) using a test point in each interval.

For example, in the interval ( 0 \leq x \leq \frac{\pi}{2} ), choose ( x = 0 ), ( y'(0) = 1 ), which is positive, indicating that ( y ) is increasing on this interval.

Similarly, test the other intervals to find where the function is increasing or decreasing.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you determine whether Rolle's theorem can be applied to #f(x) = (x^2 - 1) / x# on the closed interval [-1,1]?
- What are the local extrema, if any, of #f (x) =x/(-12x+2#?
- How do you verify that the function #f(x) = (x)/(x+2)# satisfies the hypotheses of the Mean Value Theorem on the given interval [1,4], then find all numbers c that satisfy the conclusion of the Mean Value Theorem?
- What are the absolute extrema of #f(x)=(x^2 - 1)^3 in[-oo,oo]#?
- How do you determine all values of c that satisfy the mean value theorem on the interval [0,1] for # sqrt (x(1-x))#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7