Let f:[0,3]rightarrow A$f:[0,3]\rightarrow A$ be defined by f(x)=2x^3-15x^2+36x+7$f(x)=2x^{3}-15x^{2}+36x+7$ and g:[0,infty)rightarrow B$g:[0,\infty)\rightarrow B$ be defined by g(x)=fracx^2025x^2025+1$g(x)=\frac{x^{2025}}{x^{2025}+1}$. If both the functions are onto and S=\xin Z:xin Atext or xin B\$S=\{x\in Z:x\in A\text{ or }x\in B\}$, then n(S)$n(S)$ is equal to:

### Related Formula For a function to be onto, its Codomain must equal its Range. ### Core Logic First, find range A$A$ for f(x) = 2x^3 - 15x^2 + 36x + 7$f(x) = 2x^3 - 15x^2 + 36x + 7$ over [0, 3]$[0, 3]$. Differentiating f(x)$f(x)$: f'(x) = 6x^2 - 30x + 36 = 6(x^2 - 5x + 6) = 6(x-2)(x-3)$$f'(x) = 6x^2 - 30x + 36 = 6(x^2 - 5x + 6) = 6(x-2)(x-3)$$ Critical points are x=2$x=2$ and x=3$x=3$. Evaluate f(x)$f(x)$ at boundary and critical points: - f(0) = 7$f(0) = 7$ - f(2) = 2(8) - 15(4) + 36(2) + 7 = 16 - 60 + 72 + 7 = 35$f(2) = 2(8) - 15(4) + 36(2) + 7 = 16 - 60 + 72 + 7 = 35$ - f(3) = 2(27) - 15(9) + 36(3) + 7 = 54 - 135 + 108 + 7 = 34$f(3) = 2(27) - 15(9) + 36(3) + 7 = 54 - 135 + 108 + 7 = 34$ Thus, Range A = [7, 35]$A = [7, 35]$. ### Step 1: Find Range B for g(x) Now look at g(x) = fracx^2025x^2025+1 = 1 - frac1x^2025+1$g(x) = \frac{x^{2025}}{x^{2025}+1} = 1 - \frac{1}{x^{2025}+1}$ over [0, infty)$[0, \infty)$. - At x = 0$x = 0$, g(0) = 0$g(0) = 0$. - As x to infty$x \to \infty$, g(x) to 1$g(x) \to 1$. Since g(x)$g(x)$ is continuous and monotonically strictly increasing, Range B = [0, 1)$B = [0, 1)$. ### Step 2: Find the Integer Count of Union Set S S = \x in mathbbZ : x in A text or x in B\ = mathbbZ cap (A cup B)$S = \{x \in \mathbb{Z} : x \in A \text{ or } x \in B\} = \mathbb{Z} \cap (A \cup B)$ A cup B = [0, 1) cup [7, 35]$$A \cup B = [0, 1) \cup [7, 35]$$ The integers in this set are: - From [0, 1)$[0, 1)$: x = 0$x = 0$ - From [7, 35]$[7, 35]$: x = 7, 8, 9, dots, 35$x = 7, 8, 9, \dots, 35$ Total number of integers n(S)$n(S)$: n(S) = 1 + (35 - 7 + 1) = 1 + 29 = 30$$n(S) = 1 + (35 - 7 + 1) = 1 + 29 = 30$$ ### Pattern Recognition The condition 'or' means union. Be careful not to include integers between 1$1$ and 6$6$ since they are not present in either continuous range segment. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Functions Class 12 Mathematics: Application of Derivatives