Is $f (x) = cos (x + \frac{π}{4})$ $f(x)= cos(x+(pi)/4)$ increasing or decreasing at $x = \frac{π}{3}$ $x=pi/3$ ?

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Answer 1 · 2016-11-05T09:16:59

$f (x) = cos (x + \frac{π}{4})$ $f(x)=cos(x+pi/4)$ is decreasing at $x = \frac{π}{3}$ $x=pi/3$

Whether a function is increasing or decreasing at a point depends on the value of its derivative at that point.

If derivative is positive, function is increasing and if it is negative, function is decreasing.

As $f (x) = cos (x + \frac{π}{4})$ $f(x)=cos(x+pi/4)$ , $f'(x)=-sin(x+pi/4)$ .

and at $x=pi/3$ , $f'(pi/3)=-sin(pi/3+pi/4)=-sin((7pi)/12)=-sin(pi-(5pi)/12)=-sin((5pi)/12)$ .

As $sin((5pi)/12)>0$ ( $(5pi)/12$ being acute angle), $f'(pi/3)<0$

Hence, $f(x)=cos(x+pi/4)$ is decreasing at $x=pi/3$

Is f(x)= cos(x+(pi)/4) f(x)=cos(x+π4)f(x)= cos(x+(pi)/4) increasing or decreasing at x=pi/3 x=π3x=pi/3 ?