How do you determine whether the function #f(x)= sinx-cosx# is concave up or concave down and its intervals?

Answer 1

See the explanation section.

#f(x)= sinx-cosx#
#f'(x)= cosx+sinx#
#f''(x)= -sinx+cosx#
#f''(x) = 0# where #sinx = cos x# or #tanx=1#
This happens at #x=pi/4 + pik# for integer #k#.
For #pi/4 < x < (5pi)/4# we have #sinx > cos x# so #f''(x) <0# and the graph of #f# is concave down.
For #(-5pi)/4 < x < pi/4# we have #sinx < cos x# so #f''(x) > 0# and the graph of #f# is concave up.
Both sine and cosine are periodic with period #2pi#, so
on intervals of the form #(pi/4+2pik, (5pi)/4+2pik)#, where #k# is an integer, the graph of #f# is concave down.
on intervals of the form #((-5pi)/4+2pik, pi/4+2pik)#, where #k# is an integer, the graph of #f# is concave up.

There are, of course other ways to write the intervals.

Sign up to view the whole answer

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

Sign up with email
Answer 2

To determine whether the function ( f(x) = \sin(x) - \cos(x) ) is concave up or concave down and its intervals, you need to find the second derivative of the function and then examine its sign.

  1. Find the first derivative: [ f'(x) = \frac{d}{dx} (\sin(x) - \cos(x)) = \cos(x) + \sin(x) ]

  2. Find the second derivative: [ f''(x) = \frac{d^2}{dx^2} (\cos(x) + \sin(x)) = -\sin(x) + \cos(x) ]

  3. Determine the intervals where ( f''(x) > 0 ) for concave up and where ( f''(x) < 0 ) for concave down.

  4. Solve ( -\sin(x) + \cos(x) > 0 ) for concave up: [ -\sin(x) + \cos(x) > 0 ] [ \cos(x) > \sin(x) ]

  5. Analyze the sign of ( \cos(x) - \sin(x) ) within the period ( 0 \leq x < 2\pi ) or ( 0^\circ \leq x < 360^\circ ) to determine the intervals where the function is concave up.

  6. Repeat the process to determine the intervals where the function is concave down by analyzing where ( f''(x) < 0 ).

By following these steps, you can identify the intervals where the function ( f(x) = \sin(x) - \cos(x) ) is concave up or concave down.

Sign up to view the whole answer

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

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7