How many solutions does the equation have?
cos (x/7) = 10.5x
Any help is appreciated!
What equation? The equation y= 10.5 x is, of course, a straight line. The graph of y= x/7 is not but close to x= 0 can be approximated by the horizontal straight line y= 1 so, yes, 1/10.5 is a good approximation to the one point at which the two graphs cross.
$\displaystyle cos(0)=1,\ cos\left(\frac{3.5{\pi}}{7}\right)=0$
$\displaystyle (10.5)(0)=0,\ (10.5)(3.5{\pi})>0$
The line intersects the curve after x=0.
The line $\displaystyle y=10.5x$ passes through the origin $\displaystyle (0,0)$ and has a slope of 10.5.
A line $\displaystyle y=mx$ will touch the graph of $\displaystyle y=cos\left(\frac{x}{7}\right)$ more than once, when it is tangential to it.
$\displaystyle \frac{d}{dx}cos\left(\frac{x}{7}\right)=-\frac{1}{7}sin\left(\frac{x}{7}\right)$
The derivative of the curve is the slope of the tangent.
The maximum value of the derivative is $\displaystyle \frac{1}{7}$ and the minimum value is $\displaystyle -\frac{1}{7}$
This means that lines through the origin with slopes greater than or less than these cannot touch the curve more than once.
Therefore the given line cuts the curve only once.