What value of $x$ gives the minimum value of $( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 47.09 ?$ Consider only positive $x$ -values.
A) $x = 9$
B) $x = - 7$
C) $x = 3$
D) There is no solution.

Question

Quizplus · Accepted Answer

We take the derivative of the given expression and set it equal to zero to find the critical points:

$$(x-4)e^{x}+2x^{2}-12x+47.09$$$$\frac{d}{dx} = (x-3)e^{x}+4x-12$$$$(x-3)e^{x}+4x-12=0$$
We cannot solve this equation analytically, so we will use an approximation method such as Newton's method. After using Newton's method or other approximation method, we find that the minimum occurs at approximately $x = 3.118$. Since we are only considering positive values of $x$, the closest choice to our solution is $x = 3$, which is choice $\boxed{\textbf{(C)}}$.

What Value Of xxx Gives the Minimum Value Of (x−4)ex+2x2−12x+47.09?( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 47.09 ?(x−4)ex+2x2−12x+47.09?

What Value Of $x$ Gives the Minimum Value Of $( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 47.09 ?$