Finding parabola coefficients and revenue function derivative

[tink]

Ok here are the questions...
1)Find the Parabola with equation y=ax^2 +bx +c that has slope 4 at x=1, slope -8 at x=-1 and passes through the point (2,15)


2)A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabris that is sold is a function of the selling price p, so we can write q=f(p). Then the total revenue earned with selling prince p is R(p)=pf(p).
a) What does it mean to say that f(20)= 10 000 and f`(20)= -350 (f prime) ?
b) Assuming the values in part (a), find R`(20) and interpret your answer.


I need help!

thanks

 

[Pyrrhus]

For the first problem

Use what you are given if the parabole goes through the point (2,15)
then when its x=2. its y = 15. You know the slope is the first derivative right?, make your system of equations.

For the second problem,

Understand what the first derivative means.

 

[wais]

for reaching out for help! Let's break down these two differentiation rule questions to make them a bit easier to understand and solve.

1) Find the Parabola with equation y=ax^2 +bx +c that has slope 4 at x=1, slope -8 at x=-1 and passes through the point (2,15)

To find the equation of a parabola, we need to use the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept. In this case, we are given the slopes at two points, (1,4) and (-1,-8), and a point that the parabola passes through, (2,15).

To find the slope-intercept form of the parabola, we need to use the point-slope form: y - y1 = m(x - x1). Plugging in the values, we get:

For (1,4): y - 4 = 4(x - 1)
For (-1,-8): y + 8 = -8(x + 1)

Simplifying these equations, we get:
y = 4x
y = -8x - 16

Now, we can set these two equations equal to each other to find the x-coordinate of the vertex (where the slopes are equal):
4x = -8x - 16
12x = -16
x = -4/3

To find the y-coordinate of the vertex, we plug in x = -4/3 into one of the equations:
y = 4(-4/3)
y = -16/3

So, the vertex is at (-4/3, -16/3). Now, we can plug this back into one of the equations to find the value of a:
-16/3 = a(-4/3)^2 + b(-4/3) + c
-16/3 = 16a/9 - 4b/3 + c
c = -16/3 + 16a/9 + 4b/3
c = 16a/9 + 4b/3 - 48/9
c = 16a/9 + 4b/3 - 16/3
c = (16a + 4b - 16)/3

Now, we can use the point (2,