Provide an appropriate response.
-$\text { Find the absolute maximum and minimum values of } f ( x ) = 2 x - e ^ { x } \text { on } [ 0,1 ] \text {. }$
A) Maximum = ln 4 - 2 at x = ln 2, minimum = 1 at x = 0
B) Maximum = ln 2 - 2 at x = ln 2, minimum = -1 at x = 0
C) Maximum = 0 at x = ln 2, minimum = 1 at x = 0
D) Maximum = ln 4 - 2 at x = ln 2, minimum = -1 at x = 0

Question

Quizplus · Accepted Answer

To find the absolute maximum and minimum values of $f(x) = 2x - e^x$ on the interval $[0,1]$, first, find the derivative $f'(x) = 2 - e^x$. Setting $f'(x) = 0$ gives $e^x = 2$, or $x = \ln(2)$, which is within the interval $[0,1]$. Evaluating $f(x)$ at the critical point and the endpoints of the interval:- At $x = 0$, $f(0) = 2(0) - e^0 = -1$.- At $x = \ln(2)$, $f(\ln(2)) = 2\ln(2) - e^{\ln(2)} = 2\ln(2) - 2 = \ln(4) - 2$.- At $x = 1$, $f(1) = 2(1) - e^1 = 2 - e$, which is less than $f(0)$ and $f(\ln(2))$.Therefore, the maximum value is $\ln(4) - 2$ at $x = \ln(2)$, and the minimum value is $-1$ at $x = 0$.

Provide an Appropriate Response Find the absolute maximum and minimum values of f(x)=2x−ex on [0,1]. \text { Find the absolute maximum and minimum values of } f ( x ) = 2 x - e ^ { x } \text { on } [ 0,1 ] \text {. } Find the absolute maximum and minimum values of f(x)=2x−ex on [0,1].

Provide an Appropriate Response $\text { Find the absolute maximum and minimum values of } f ( x ) = 2 x - e ^ { x } \text { on } [ 0,1 ] \text {. }$