Area bounded by f(x) = $x^3/3$ - $x^2$ +a and the straight line x=0, x=2 and the x axis is minimum. Find a

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This is the screen shot of the function when a=0.6.

Scrolling the value of "a" from 0 to 9, I am finding the function to have least value between 0.6 to 0.7...How to prove this mathematically?

  • $\begingroup$ Do you have any ideas about how to calculate the area mathematically? $\endgroup$ – Minus One-Twelfth Feb 15 at 5:00
  • $\begingroup$ Yes. I believe it comes from integration of bounded functions $\endgroup$ – Scáthach Feb 15 at 5:02
  • $\begingroup$ But with "a"as a parameter, how will it help us? $\endgroup$ – Scáthach Feb 15 at 5:03
  • $\begingroup$ Correct! We can use integration. Maybe you can try using integration to find the area as a function of $a$, and then investigate this function. $\endgroup$ – Minus One-Twelfth Feb 15 at 5:04
  • $\begingroup$ Okay. Let me try then $\endgroup$ – Scáthach Feb 15 at 5:06

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