Question (a). Apply the second derivative test to find the relative extrema of t

Question

Question

(a). Apply the second derivative test to find the relative extrema of the function f(x)=x3-3x2+1.

(b). Apply the first derivative test to find the interval where the function

g(t)=t4-4t3-4 is increasing and decreasing, as well as the relative extrema.

Question

(a). Find two positive numbers such that their product is 75, and so that the first plus three times the second is a minimum.

(b). A rectangular page is to contain 18 square inches of print. The margins at the top and bottom are 0.5 inches and on each side are 1 inch. What dimensions minimizes the amount of paper used.

Explanation / Answer

a)f(x) = x^3-3x^2 +1,
for extremum
f`(x) = 3x^2-6x=0
so x= 0 or x=2
now,
f``(x) =6x-6 ,
f``(0)=-6<0 ,
hence , x=0 is a point of maxima
similarly,
f``(2)=6>0
hence , x=2 is a point of minima

b) g(t) = t^4-4t^3-4,
g'(t) = 4t^3-12t^2= 4 *t^2(t-3)
it is positve for x>3 and negative for x<3
hence,
g(t) is increasing for (3, infinity)
g(t) is decreasing for (- infinity,3)

c) let x and y be the numbers,
xy = 75 .......(1)
x+3y is a mimimum,
so , x= 3y ,..........(2)
From(1) AND (2)
3 y^2 = 75 ,
y = 5 and x= 3*Y = 3*5 = 15

d)

let l and b be dimension,
(l-1) * (b-2) = 18 ,

we need to minimise the paper used,
so ,
l-1 = b-2,
l = b-1...,

(b-1)^2 = 18,
b = 4.2426 +1 = 5.2426 ,

and l = 4.2426 inch

Related Questions

Navigate

• Browse (All)
• Browse Q
• Subjects