Let P(x) be a polynomial of degree 3 such that it satisfies the...

Let the cubic polynomial be $P(x) = ax^3 + bx^2 + cx + d$.

The derivatives are:
$P'(x) = 3ax^2 + 2bx + c$
$P''(x) = 6ax + 2b$

Adding these gives:
$P(x) + P'(x) + P''(x) = ax^3 + (b + 3a)x^2 + (c + 2b + 6a)x + (d + c + 2b)$

We are given that this sum equals $x^3 + 5x^2 + 13x + 11$. Equating the corresponding coefficients:

Coefficient of $x^3$: $a = 1$

Coefficient of $x^2$: $b + 3a = 5 \Rightarrow b + 3(1) = 5 \Rightarrow b = 2$

Coefficient of $x$: $c + 2b + 6a = 13 \Rightarrow c + 2(2) + 6(1) = 13 \Rightarrow c + 10 = 13 \Rightarrow c = 3$

Constant term: $d + c + 2b = 11 \Rightarrow d + 3 + 2(2) = 11 \Rightarrow d + 7 = 11 \Rightarrow d = 4$

Thus, the polynomial is $P(x) = x^3 + 2x^2 + 3x + 4$.

We need to find $P(2)$:
$P(2) = (2)^3 + 2(2)^2 + 3(2) + 4 = 8 + 8 + 6 + 4 = 26$.

Answer: 26