Given f(x) = e^{sin x} + e^{cos x}. The globa | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given the function $f(x) = e^{\sin x} + e^{\cos x}$.To find the critical points, we set the derivative $f'(x) = 0$:$f'(x) = e^{\sin x} \cdot \cos

Vedclass · Accepted Answer

exists at infinitely many points

Vedclass · Answer

(C) Given the function $f(x) = e^{\sin x} + e^{\cos x}$.To find the critical points, we set the derivative $f'(x) = 0$:$f'(x) = e^{\sin x} \cdot \cos x + e^{\cos x} \cdot (-\sin x) = 0$$e^{\sin x} \cos x = e^{\cos x} \sin x$$\frac{e^{\sin x}}{e^{\cos x}} = \frac{\sin x}{\cos x}$$e^{\sin x - \cos x} = 	an x$Since $f(x)$ is a periodic function with period $2\pi$, the solutions for $x$ occur periodically.At $x = \frac{\pi}{4}$, $e^{\sin(\pi/4) - \cos(\pi/4)} = e^0 = 1$ and $	an(\pi/4) = 1$. Thus, $x = \frac{\pi}{4}$ is a critical point.Due to the periodicity of $\sin x$ and $\cos x$, the function $f(x)$ repeats its values every $2\pi$.Therefore, the global maximum occurs at $x = 2n\pi + \frac{\pi}{4}$ for all integers $n$.Since there are infinitely many such points, the global maximum exists at infinitely many points.

Given $f(x) = e^{\sin x} + e^{\cos x}$. The global maximum value of $f(x)$

Similar Questions

The two positive numbers with sum $t$,and the sum of their squares is minimum are

If the lengths of three sides of a trapezium other than the base are equal to $10 \ cm$,find the maximum area of the trapezium.

The maximum value of the function $f(x)=2x^3-15x^2+36x-48$ on the set $A=\{x | x^2+20 \leq 9x\}$ is

If $x+y=60$,$x>0$,$y>0$,then the maximum value of $x y^3$ is

If local maximum of $f(x) = \frac{ax + b}{(x - 1)(x - 4)}$ exists at $(2, -1)$, then $a + b =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module