Solved Examples on How to Find Global Maxima and Minima

Example 1: Find the global maxima and minima value of function f(x) = 2x² – 4x in the interval [0, 2].

Solution:

f(x) = 2x² – 4x

First find f'(x)

f'(x) = 4x – 4

Now put f'(x) = 0

4x – 4= 0

4x = 4

x = 1

There is only one critical point of function i.e., x = 1

Now find the value of f(x) at critical point

At x = 1 f(1) = – 2

Now find the value of f(x) at extremum points I.e.,

x =0 f(0) = 0

x = 2 f(2) = 0

So, f(x) has its global maxima value 0 at x= 0 and x = 2 and global minima value -2 at x = 1.

Example 2: Find the global maxima and minima of function f(x) = 4e^x + 3 in the interval [0, 3].

Solution:

f(x) = 4e^x + 3

First find f'(x)

f'(x) = 4e^x

The above equation will never be zero for any value of x within the interval. So, we find values of f(x) at extremum points. Now find the value of f(x) at extremum points I.e.,

x =0 f(0) = 7

x = 3 f(3) = 4e³ + 3

So, f(x) has its global maxima value 4e³ + 3 at x= 0 and global minima value 7 at x = 0.

Example 3: Find the global maximum value and minimum value of function f(x) = 1 / (x – 2) in its entire domain.

Solution:

The domain of the function f(x) = 1 / (x – 2) is R – {2}.

f'(x) = -1 / (x – 2)²

There is no point in function domain where f'(x) = 0

So, we check for the values of function when x tends to ∞

• x→-∞ ⇒ f(x) → 0
• x→∞ ⇒ f(x) → 0
• x→ 2 ⇒ f(x) → ∞

Therefore, there is no maximum value and minimum value exists when x→-∞ or ∞ ⇒ f(x) → 0.

Do you ever wonder how to find the highest or lowest points of something? Maybe you’re trying to figure out the best temperature for baking cookies or the shortest route to school. In the world of mathematics, we call these high points “global maximums” and low points “global minimums.” They’re like the peaks and valleys of a roller coaster, showing us the highest and lowest points along the ride.

In this article, we’ll take a journey into the world of finding global maximums and minimums.