Find the values of function

Question

a) Find the values of x for which the function f(x)=1+4x-x² is increasing.

 

b) The graph of y=ax³ + bx² + cx passes through the point (-1, 16) and has a stationary point at (1, -4). Find the values of     a, b and c.  

 

Answer 1

(a)

1+4x-x^2

-x^2+4x+1

-(x^2-4x-1)

-(x^2-4x+4-4-1)

-(x-2)^2+5

It is the equation of inverted parabola with vertex as (2,5)

So, function is increasing for x<2

 

(b)

 

y =ax^3+bx^2+cx

It passes through (-1,16)

So,

16=-a+b-c

It also passes through (1,-4)

-4=a+b+c

Solving the above two equations

We get b= 6

 

(1,-4)is also a stationary point

y'=3ax^2+2bx+c

y''=6ax+2b=0

x=-b/3a

  =-6/3a=-2/a=1

a=-2

 

Substituting above----

+2+6-c=16

c=-8

 

So the desired values are-----

a=-2.   b =6.    c =-8