Question

For what values of \(x\) does the graph of \(f\) have a horizontal tangent?

\(f\left( x \right) = {e^x}\cos x\)

Short answer

The required value of \(x = n\pi + \frac{\pi }{4},\,\,\,\,\,\,\,{\rm{for}}\,\,n \in {\rm I}\).

Step-by-step solution

Horizontal Tangents of Trigonometric Functions

When the derivative of any particular trigonometric function is being equated to zero, then that trigonometric function is said to have horizontal tangents along with its variable.

 Step 2: Differentiation ofthe given function and solving for \(x\)

The given function is\(f\left( x \right) = {e^x}\cos x\).

Differentiating the given functionwith respect to\(x\)and equating it to zero, we get:

\(\begin{aligned}\frac{d}{{dx}}\left( {f\left( x \right)} \right) &= 0\\\frac{d}{{dx}}\left( {{e^x}\cos x} \right) &= 0\\{e^x}\left( {\cos x - \sin x} \right) &= 0\\\cos x - \sin x &= 0\\\tan x &= 1\\x &= n\pi + \frac{\pi }{4},\,\,\,\,\,\,\,\,\,\,\,\,\forall n \in {\rm I}\end{aligned}\)

Hence, the required answer is\(x = n\pi + \frac{\pi }{4},\,\,\,\,\,\,\,{\rm{for}}\,\,n \in {\rm I}\).