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

Answer 1

See below.

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

#(d^2y)/(dx^2)f(x)>0color(white)(888)# Convex( concave up )

#(d^2y)/(dx^2)f(x)<0color(white)(888)# Concave( concave down )

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

#(d^2y)/(dx^2)f(x)=dy/dx(dy/dxf(x))#

For #f(x)=cos(x)#

#dy/dx=-sin(x)#

#(d^2y)/(dx^2)(cos(x))=dy/dx(-sin(x))=-cos(x)#

Now we solve the inequalities:

#-cos(x)>0color(white)(888)[1]#

#-cos(x)<0color(white)(888)[2]#

#-cos(x)>0color(white)(888)[1]#

#pi/2 < x< (3pi)/2#

#-cos(x)<0color(white)(888)[2]#

#0 < x < pi/2color(white)(88)# , #(3pi)/2 < x < 2pi #

Notice that #bb((3pi)/2)# is on the point where the function changes from convex to concave. This is called a point of inflection ( inflexion in the UK ), so at #bb((3pi)/2)# it is neither concave nor convex.

This is verified by its graph:

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Answer 2

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.

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Answer from HIX Tutor

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.

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.

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