Respuesta :
Answer:
The given function is
[tex]g(x)=(5-x)(2x+3)[/tex]
The x-intercepts of the graph are at [tex]g(x)=0[/tex]
[tex](5-x)(2x+3)=0[/tex]
[tex]5-x=0 \implies x=5\\2x+3=0 \implies 2x=-3 \implies x=-\frac{3}{2}[/tex]
Therefore, the x-intercepts are [tex](5,0)[/tex] and [tex](-\frac{3}{2},0)[/tex].
The midpoint can be found with the formula
[tex]P_{M}=(\frac{x_{1}+x_{2} }{2} ,\frac{y_{1} +y_{2} }{2} )[/tex]
[tex]P_{M}=(\frac{5-\frac{3}{2} }{2},0)\\P_{M}= (\frac{\frac{10-3}{2} }{2} ,0)\\P_{M}= (\frac{\frac{7}{2} }{2} ,0)\\ P_{M}= (\frac{7}{4} ,0)[/tex]
The minimum value about a parabola is at its vertex. In this case, the parabola has maxium vale only.
The vertex has coordinates of [tex]V(h,k)[/tex], where [tex]h=-\frac{b}{2a}[/tex] and [tex]k=f(h)[/tex].
Solving the product of the given expression
[tex]g(x)=(5-x)(2x+3)=10x+15-2x^{2} +-3x=-2x^{2} +7x+15[/tex]
Where [tex]a=-2[/tex], [tex]b=7[/tex] and [tex]c=15[/tex].
[tex]h=-\frac{b}{2a}=-\frac{7}{2(-2)}=\frac{7}{4}[/tex]
[tex]k=f(\frac{7}{4})=-2(\frac{7}{4} )^{2} +7(\frac{7}{4})+15 =-2(\frac{49}{16} )+\frac{49}{4}+15\\ k=-\frac{49}{8}+\frac{49}{4} +15=\frac{-49+98+120}{8} =\frac{169}{8}\\ k=\frac{169}{8}[/tex]
The x-intercept is [tex](5, 0) \ and \ (-\frac{3}{2}, 0 )[/tex]
The midpoint of the x-intercept is 1.75.
The extreme value is a minimum and the value is [tex]\frac{169}{8}[/tex]
Given the function g(x) expressed as;
[tex]g(x) = (5 - x)(2x+3)[/tex]
a) The x-intercept of the function occur at when g(x) = 0. Substitute g(x) = 0 into the function as shown:
[tex](5-x)(2x+3)=0\\5-x=0 \ and \ 2x+3 =0\\x = 5 \ and \ x=-\frac{3}{2}[/tex]
The x-intercept is [tex](5, 0) \ and \ (-\frac{3}{2}, 0 )[/tex]
b) The midpoint of the x-intercept is expressed as:
[tex]m= (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} )\\m=(\frac{5-1.5}{2}, 0)\\m=(\frac{3.5}{2},0 )\\m=(1.75, 0)[/tex]
The midpoint of the x-intercept is 1.75.
c) The extreme value occurs at g(k) = 0 where [tex]k=\frac{-b}{2a}[/tex]
Expand the function given
g(x)=(5−x)(2x+3)
g(x) = 10x + 15 - 2x² - 3x
g(x) = - 2x² + 7x + 15
b = 7 and a = -2
[tex]k=-\frac{7}{2(-2)}\\k = \frac{7}{4}[/tex]
Get [tex]g(\frac{7}{4} )[/tex]
Recall that g(x) = - 2x² + 7x + 15
[tex]g(\frac{7}{4} ) = - 2(\frac{7}{4} )^2 + 7(\frac{7}{4} ) + 15\\g(\frac{7}{4} ) =-2(\frac{49}{16} )+\frac{49}{4} +15\\g(\frac{7}{4} ) =-\frac{98}{16} +\frac{49}{4} +15\\g(\frac{7}{4} ) = \frac{169}{8}[/tex]
Hence the extreme value is a minimum and the value is [tex]\frac{169}{8}[/tex]
Learn more here: https://brainly.com/question/17207969
