suppose a function f is differentiable for all x and f(0)=0. If g(x) is defined by g(x)= f(x)cos(x), which of the following statements must be true.

1) there exist a number c in the interval (0,pi/2) such that g'(c)=0

2)there exist a number c in the interval (pi/2,pi) such that g'(c)=0

3)there exist a number c in the interval(-pi/2,0) such that g'(c)=0

Guest Feb 22, 2015

#2**+10 **

This question made me think

$$\\cos\;\frac{\pi}{2}=0,\qquad cos\;\frac{-\pi}{2}=0,\qquad and \qquad f(0)=0\\\\$$

Now since g(x)=f(x)*cosx it follows that g(x)=0 when x = -pi/2, 0 or pi/2

Hence g(x) must be at least one turning point between -pi/2 and 0 and at least one between 0 and pi/2

If you are not sure about this, plot the 3 points and think about what the graph will have to look like.

NOW when g(x) turns the gradient of the tangent must be 0 **SO** g'(x)=0

HENCE

1 and 3 must both be correct.

Melody
Feb 22, 2015

#2**+10 **

Best Answer

This question made me think

$$\\cos\;\frac{\pi}{2}=0,\qquad cos\;\frac{-\pi}{2}=0,\qquad and \qquad f(0)=0\\\\$$

Now since g(x)=f(x)*cosx it follows that g(x)=0 when x = -pi/2, 0 or pi/2

Hence g(x) must be at least one turning point between -pi/2 and 0 and at least one between 0 and pi/2

If you are not sure about this, plot the 3 points and think about what the graph will have to look like.

NOW when g(x) turns the gradient of the tangent must be 0 **SO** g'(x)=0

HENCE

1 and 3 must both be correct.

Melody
Feb 22, 2015