Question

please answer it correctly​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{\displaystyle\int(4x^3+2x)e^{x^4+x^2}\,dx=e^{f(x)}+C}$

$\sf{Put\,\,\,\,x^4+x^2=t}$

$\sf{\implies\,(4x^3+2x)dx=dt}$

So,

$\sf{\implies\,\displaystyle\int\,e^{t}\,dt=e^{f(x)}+C}$

$\sf{\implies\,e^{t}+C=e^{f(x)}+C}$

$\sf{\implies\,e^{x^4+x^2}+C=e^{f(x)}+C}$

$\sf{So,\,\,\,f(x)=x^4+x^2}$

Finding the minimum value of f(x),

$\sf{f^{\prime}(x)=4x^3+2x=0}$

$\sf{\implies\,2x(2x^2+1)=0}$

$\sf{\implies\,x=0}$

$\sf{f^{\prime\prime}(x)=12x^2+2}$

At x=0, f''(x) is positive

So, at x = 0, f(x) attains its minimum value

So,

Minimunm value of f(x)=0