What value of $$x$$ gives the minimum value of $( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 39.09 ?$ Consider only positive $x$ -values.
A) $x = 1$
B) $x = - 15$
C) $x = 3$
D) There is no solution.

Question

Quizplus · Accepted Answer

To find the minimum value of the given expression, we need to find the critical points. 
Taking the derivative with respect to $x$ and equating it to zero, we get 
$$e^x(x-3)+4x-12=0$$
As we need to consider only positive values of $x$, we can eliminate the root $x=3$ as it doesn't satisfy the constraint. 
Now, we need to check the values of the expression at the endpoints. Taking $x=0$ and $x\rightarrow \infty$, we get 
$$(x-4)e^x+39.09=35.09\text{ (approx.)}$$ 
and 
$$\lim_{x\rightarrow \infty}(x-4)e^x+2x^2-12x+39.09=\infty$$ 
Hence, the minimum value is achieved at the critical point $x=1$ and the answer is $\textbf{(A)}$.

What Value Of xxx Gives the Minimum Value Of (x−4)ex+2x2−12x+39.09?( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 39.09 ?(x−4)ex+2x2−12x+39.09?

What Value Of $x$ Gives the Minimum Value Of $( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 39.09 ?$