Question

Given f(x) = e^sinx+e^cosx, The global maximum value of f(x)

• A. does not exist
• B. exist at a point in(0,\(\frac{\pi}{2}\)) and its value is 2e\(^{\frac{1}{\sqrt2}}\).
• C. exists at infinitely many points
• D. exists at x=0 only

Answer 1

The Correct Option is C

Step 1: Let $h(x) = f(x) - g(x)$ 

Step 2: Since both $f(x)$ and $g(x)$ are differentiable functions, their difference $h(x)$ is also differentiable. 

Step 3: To find maximum or minimum values of $h(x)$, we find critical points by setting:
$h'(x) = 0$ 

Step 4: Differentiate $h(x)$: 
Suppose $f(x)$ and $g(x)$ are such that:
$h'(x) = f'(x) - g'(x)$ 

Step 5: Solve the equation:
$h'(x) = 0 \Rightarrow f'(x) = g'(x)$ 

Step 6: Depending on the functions $f$ and $g$, this equation might have infinitely many solutions. For example, if $f'(x)$ and $g'(x)$ are trigonometric functions that intersect periodically, $h'(x) = 0$ at infinitely many values of $x$. 

Step 7: Therefore, $h(x)$ can attain a maximum value at infinitely many points. 

Final Answer: (C) exists at infinitely many points