Let $f(x) = ax^7 + bx^3 - cx - 5$ where $a, b, c$ are constant, if $f(-7) = 7$, then $f(7)$ is equal to-

Explanation

$$ \begin{aligned} f(x) &= ax^7 + bx^3 - cx - 5 \\ f(-7) &= a(-7)^7 + b(-7)^3 - c(-7) - 5 \\ +7 &= -a(7)^7 - b(7)^3 + 7c - 5 \quad (\because f(-7) = +7) \\ 7 &= -\{a(7)^7 + b(7)^3 - 7c\} - 5 \\ 12 &= -\{a(7)^7 + b(7)^3 - 7c - 5 + 5\} \\ 12 &= -\{f(7) + 5\} \\ 12 &= -f(7) - 5 \\ \therefore f(7) &= -17 \end{aligned} $$ ## Page 35 ## Layout Attribution (Critical) A table of layout regions detected on this page is provided inside <layout_regions> tags. Each region has an id, a label (e.g. text, table, image, header, footer, chart, etc.), and a bounding box in box_2d format [y_min, x_min, y_max, x_max] normalized to 0..1000 with top-left origin. Your task: before EVERY markdown element you output (paragraph, heading, table, list, code block, mermaid block, image placeholder, equation, page_header, page_footer, etc.), insert an HTML comment listing the layout region id(s) that correspond to that element: ``` ``` Rules: - Every markdown element MUST be preceded by a comment. - A single element may map to multiple layout regions (e.g. a table spanning two detected regions). List all matching ids separated by spaces. - A layout region should appear in at most one comment. - Use the bounding box and label to match visual regions on the screenshot to the content you are transcribing. - If a visible region has no matching layout id, you can omit the attribution comment for that element. - The layout region table is a reference for attribution only — the OCR text content is provided…