What value of $x$ gives the minimum value of $( x - 6 ) e ^ { x } + 3 x ^ { 2 } - 30 x + 225.41 ?$ Consider only positive $x$ -values.
A) $$x = 2$$
B) $$x = - 25$$
C) $x = 5$
D) There is no solution.

Question

Quizplus · Accepted Answer

To find the minimum value, we need to take the derivative of the expression and set it equal to zero. We get:

$$(x-5)e^x+3x(x-10)$$
$$(x-5)e^x+3x^2-30x$$

Setting this equal to zero and solving for x gives us:

$$x=5$$

To confirm that this is indeed the minimum value, we can take the second derivative:

$$(x-6)e^x+6x-30$$

Plugging in x=5 gives us 0, which means the function is concave up at x=5, and hence this is the minimum value.

What Value Of xxx Gives the Minimum Value Of (x−6)ex+3x2−30x+225.41?( x - 6 ) e ^ { x } + 3 x ^ { 2 } - 30 x + 225.41 ?(x−6)ex+3x2−30x+225.41?

What Value Of $x$ Gives the Minimum Value Of $( x - 6 ) e ^ { x } + 3 x ^ { 2 } - 30 x + 225.41 ?$