# Is #f(x)=cosx# concave or convex at #x=(3pi)/2#?

See below.

We can determine where a function is convex or concave, by using the second derivative. If:

The second derivative is just the derivative of the first derivative. .i.e.

For

Now we solve the inequalities:

Notice that

This is verified by its graph:

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

To determine if ( f(x) = \cos(x) ) is concave or convex at ( x = \frac{3\pi}{2} ), we need to check the second derivative of the function at that point.

First, find the second derivative of ( f(x) = \cos(x) ):

[ f'(x) = -\sin(x) ] [ f''(x) = -\cos(x) ]

Now, evaluate the second derivative at ( x = \frac{3\pi}{2} ):

[ f''\left(\frac{3\pi}{2}\right) = -\cos\left(\frac{3\pi}{2}\right) ]

Since the cosine function is negative at ( \frac{3\pi}{2} ), ( f''\left(\frac{3\pi}{2}\right) ) is negative.

A function is concave if its second derivative is negative, and convex if its second derivative is positive. Therefore, at ( x = \frac{3\pi}{2} ), the function ( f(x) = \cos(x) ) is concave.

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 find the exact relative maximum and minimum of the polynomial function of #f(x) = x^3 - 6x^2 + 9x +1#?
- What are the points of inflection, if any, of #f(x)=2x^3 -10x^2 +3 #?
- On what intervals the following equation is concave up, concave down and where it's inflection point is (x,y) #f(x)=x^8(ln(x))#?
- What are the points of inflection, if any, of #f(x)=x^4-x^2+5 #?
- How do you find the exact relative maximum and minimum of the polynomial function of #y=x^2 +x -1#?

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