1. Find the value of c if the function f(x) = c ln(x) - x^2 has a critical point

ID: 2839477 • Letter: 1

Question

1. Find the value of c if the function f(x) = c ln(x) - x^2 has a critical point at the value x = 4.

Is this critical point a local max, local min, or neither?

2. Let f(x) = ax^2 ln(x)+bx^3. Find the values of a and b if it is known that f'(1) = 14 and f''(1) = 30.

3. Given the function below, find where f is increasing, where f is decreasing, where f is concave up, where

f is concave down, and all critical points of f

f(x) = 2x^3 + 3x^2 - 36x + 7

I need to understand how to solve these problems, so please explain clearly your answers. I will give away points asap

f(x) = c ln(x) - x^2

f'=c/x-2x=0

c/x=2x

x^2=c/2

x=sqrt(c/2)

x=4

16=c/2

c=32

f(x) = ax^2 ln(x)+bx^3

f'=2axln(x)+axln(x)+3bx^2

f'(1) = 14

f'=2aln(1)+aln(1)+3b=14

3b=14

b=14/3

f'=2axln(x)+axln(x)+3bx^2

f''=2aln(x)+2a+6bx=30

f''=2aln(1)+2a+6b=30

2a+6b=30

2a+6(14/3)=30

2a+28=30

a=1

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