I looked up other questions regarding asymptotic notations but none of them are similar to this.
Here is the equation given:
10 n^3 + 15 n^4 + 100 n^2 x 2^n = O(n^2 x 2^n)
I need to find the appropriate c and n0.
What I have done so far:
10 n/2^n + 15 n^2/2^n + 100 <= c (Dividing by dominant after writing the definition of Big - Oh)
After finding the maxima of 10 n/2^n + 15 n^2/2^n by differentiating wrt n (overkill?) and found that n = 3
And the required constant c=121 after plugging in n = 3 in the above equation.
Whatever I have done, is it correct?
Also, would it be wrong to claim an answer of c=125 and n0=1?
Assuming n > 0 (nk is notation for n^k for the sake of readability)
10n3 + 15n4 + 100n2 2^n <= c n2 2^n
<=> 10n3 + 15n4 <= (c - 100) n2 2^n
<=> 10n + 15n2 <= (c - 100) 2^n
<=> c >= (10n + 15n2) / 2^n + 100
Replace n with desired n0 >= argmax((10n + 15n2) / 2^n + 100) here. e.g. with n0 = 3
c >= (10 * 3 + 15 * 3^2) / 2^3 + 100 = 120.625