What is the extrema of the following function?

Question

Find the extrema of the following function on the given interval, if there are any, and determine the values of 𝑥 at which the extrema occur.

a. 𝑓(𝑥) = 1/𝑥 on [−2 , 3]

b. 𝑓(𝑥) = 4𝑥³ − 3𝑥² − 6𝑥 + 3 on [−1 , 10]

c. 𝑓(𝑥) = 2 cos 𝑥 on [− 2𝜋/3 , 𝜋/3 ]

d. 𝑓(𝑥) = 𝑒^𝑥/𝑥 on [ 1/2 , 2]

Answer 1

The process to find the extrema of a function is to first take the derivative of the function, set that to zero and solve for x. Then you need to also check the interval end points against the original function to check for minimum and maximum values.

a) f(x) = 1/x

f'(x) = -1/x², solving for x gives ±∞ which is not in [-2,3].

f(-2) = -1/2, f(3) = 1/3 which are the extrema.

b) f(x) = 4x³-3x²-6x+3

f'(x) = 12x²-6x-6 --> x = -1/2, 1 which are in the interval

f(-1/2) = 19/4

f(1) = 4-3-6+3 = 10

f(-1) = -4-3+6+3 = 2

f(10) = 4000-300-60+3 = 3643

c) f(x) = 2cos(x)

f'(x) = -2sin(x) --> x = 0, π --> π is not in the interval

f(0) = 2

f(-2π/3) = 2cos(-2π/3) = 2(-1/2) = -1

f(π/3) = 2cos(π/3) = 2(1/2) = 1

d) f(x) = e^x/x on [1/2,2]

f'(x) = (xe^x - e^x)/x² = e^x(x-1)/x² --> x = -∞, 1 (-∞ not in [1/2,2])

f(1) = e ≈ 2.7

f(1/2) = 2e^1/2 ≈ 3.3

f(2) = e²/2 ≈ 3.7