The minimum value of $$e^{(x^{4} - x^{3} + x^{2})}$$​ is:

Correct option is B

Given:
​$$y = e^{(x^{4} - x^{3} + x^{2})}$$​​
Formula used:
The minimum value of $$e^{g(x)}$$​ occurs when g(x) is minimum.
Solution:
Let $$f(x) = x^{4} - x^{3} + x^{2}$$​​
Differentiate:
​$$f'(x) = 4x^{3} - 3x^{2} + 2x\\= x(4x^{2} - 3x + 2)$$​​
The quadratic $$4x^{2} - 3x + 2$$​ has discriminant:
​$$D = (-3)^{2} - 4(4)(2) = 9 - 32 < 0$$​​
So,
$$4x^{2} - 3x + 2 > 0$$​ for all x
Hence,
$$f'(x) = 0$$​ only at x = 0
Second derivative:
​$$f''(x) = 12x^{2} - 6x + 2$$​​
At x = 0:
​$$f''(0) = 2 > 0$$​​
So f(x) has a minimum at x = 0.
Minimum value of f(x):
​$$f(0) = 0$$​​
Therefore,
Minimum value of $$e^{(x^{4} - x^{3} + x^{2})}$$​​
​$$= e^{0}\\= 1$$​​
The correct answer is (b) 1.