Q.36 to 65 (Two marks Questions)Q.Consider the function f(x) = 2x3-3x2 in the domain [-1, 2]. The global minimum of f(x) is ____(Important - Enter only the numerical value in the answer)Correct answer is '-5'. Can you explain this answer?

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Answer

Global Minimum of f(x) = 2x^3 - 3x^2 in the domain [-1, 2]

To find the global minimum of the given function f(x) = 2x^3 - 3x^2 in the domain [-1, 2], we need to analyze the critical points and endpoints of the function within the given domain.

Finding the critical points:

To find the critical points, we need to find the values of x where the derivative of the function is equal to zero or undefined.

The derivative of f(x) is obtained by differentiating each term of the function with respect to x:

f'(x) = d/dx (2x^3) - d/dx (3x^2)
= 6x^2 - 6x

Next, we set f'(x) = 0 and solve for x:

6x^2 - 6x = 0
6x(x - 1) = 0

So, the critical points are x = 0 and x = 1.

Finding the endpoints:

The given domain is [-1, 2], which means we need to evaluate the function at x = -1 and x = 2 as well.

f(-1) = 2(-1)^3 - 3(-1)^2
= -2 - 3
= -5

f(2) = 2(2)^3 - 3(2)^2
= 16 - 12
= 4

Comparing the values:

Now, we compare the values of the critical points and endpoints to find the global minimum.

f(0) = 2(0)^3 - 3(0)^2
= 0

f(1) = 2(1)^3 - 3(1)^2
= 2 - 3
= -1

We can see that the minimum value of f(x) within the given domain is -5, which occurs at x = -1.

Therefore, the global minimum of f(x) = 2x^3 - 3x^2 in the domain [-1, 2] is -5.

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