Question

Find the critical point of the following problems
a. F(x)=8-3x+x^2
b. F(x)=x^4-18x^2+9
c. F(x)=x^3-5x^2-8x+3
d. F(x)= x^2/(x^2+1)
e F(x)=x^2/(x^2-1)
f F(x)=(x^2-1)^3

Answer 1

a)

$F(x)=8-3x+x^2\\ F'(x)=-3+2x\\ \text{At critical point, derivative is 0}\\ -3+2x=0\\ 2x=3\\ x=1.5\\ F(x)=8-3(1.5)+(1.5)^2=5.75\\ \text{Critical point is (1.5,5.75)}$

b)

$F(x)=x^4-18x^2+9\\ F'(x)=4x^3-36x\\ 4x^3-36x=0\\ x=0,3\\ \text{at x=0, F(x)=9}\\ \text{at x=3, F(x)=-72}\\ \text{Critical points are (0,9) and (3,-72)}$

c)

$F(x)=x^3-5x^2-8x+3\\ F'(x)=3x^2-10x-8\\ 3x^2-10x-8=0\\ x=4,\frac{-2}{3}\\ \text{At x=4, F(x)=-2}\\ \text{At x=}\frac{-2}{3}\text{ F(x)=}\frac{157}{27}\\ \text{Critical points are (4,-2) and } (\frac{-2}3,\frac{157}{27})\\ d)\\ F(x)=\dfrac{x^2}{x^2+1}\\ \text{has no critical point}\\ e)F(x)=\dfrac{x^2}{x^2-1}\\ \text{has no critical point}\\ f) F(x)=(x^2-1)^3\\ F'(x)=2x(x^2-1)^2\\ 2x(x^2-1)^2=0\\ x=0,-1,1\\ \text{critical points are (0,-1),(-1,0) and (1,0)}$