Let f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3)$f(x) = x^{3} + x^{2} f'(1) + 2x f''(2) + f'''(3)$, x in R$x \in R$. Then the value of f'(5)$f'(5)$ is :

### Related Formula fracddx (x^n) = nx^n-1$$\frac{d}{dx} (x^n) = nx^{n-1}$$ ### Core Logic Differentiate the given polynomial function iteratively to find expressions for f'(x)$f'(x)$ and f''(x)$f''(x)$, treating f'(1)$f'(1)$, f''(2)$f''(2)$, and f'''(3)$f'''(3)$ as constant values. ### Step 1: First and Second Derivatives f'(x) = 3x^2 + 2x f'(1) + 2f''(2)$$f'(x) = 3x^2 + 2x f'(1) + 2f''(2)$$ f''(x) = 6x + 2f'(1)$$f''(x) = 6x + 2f'(1)$$ Substitute x = 2$x = 2$ into the second derivative to create a relation: f''(2) = 12 + 2f'(1)$$f''(2) = 12 + 2f'(1)$$ ### Step 2: Substitute and Solve for Constants Substitute f''(2)$f''(2)$ back into f'(x)$f'(x)$: f'(x) = 3x^2 + 2x f'(1) + 2(12 + 2f'(1))$$f'(x) = 3x^2 + 2x f'(1) + 2(12 + 2f'(1))$$ f'(x) = 3x^2 + 2(x + 2)f'(1) + 24$$f'(x) = 3x^2 + 2(x + 2)f'(1) + 24$$ Now put x = 1$x = 1$ to solve for f'(1)$f'(1)$: f'(1) = 3(1)^2 + 2(1 + 2)f'(1) + 24$$f'(1) = 3(1)^2 + 2(1 + 2)f'(1) + 24$$ f'(1) = 3 + 6f'(1) + 24$$f'(1) = 3 + 6f'(1) + 24$$ -5f'(1) = 27 implies f'(1) = -frac275$$-5f'(1) = 27 \implies f'(1) = -\frac{27}{5}$$ ### Step 3: Calculate Required Value Determine the exact form of f'(x)$f'(x)$: f'(x) = 3x^2 + 2(x + 2)left(-frac275right) + 24$$f'(x) = 3x^2 + 2(x + 2)\left(-\frac{27}{5}\right) + 24$$ f'(x) = 3x^2 - frac545x - frac1085 + frac1205$$f'(x) = 3x^2 - \frac{54}{5}x - \frac{108}{5} + \frac{120}{5}$$ f'(x) = 3x^2 - frac545x + frac125$$f'(x) = 3x^2 - \frac{54}{5}x + \frac{12}{5}$$ Substitute x = 5$x = 5$ to find f'(5)$f'(5)$: f'(5) = 3(25) - frac545(5) + frac125$$f'(5) = 3(25) - \frac{54}{5}(5) + \frac{12}{5}$$ f'(5) = 75 - 54 + frac125 = 21 + frac125 = frac105 + 125 = frac1175$$f'(5) = 75 - 54 + \frac{12}{5} = 21 + \frac{12}{5} = \frac{105 + 12}{5} = \frac{117}{5}$$ ### Pattern Recognition Treat derivatives evaluated at specific points (like f'(1)$f'(1)$) as fixed scalar constants. Substitute values back sequentially to solve the linear system of constants. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Method of Differentiation