ShapeofGraphPtI - 3-masala yechimi | MathModel.uz

4.5-bo'lim : Grafik shakli, I qism

3. Quyida ning grafigi keltirilgan hosila funktsiyaning. Ushbu grafikdan qaysi intervallarni aniqlang funktsiyasi ortadi va kamayadi.

$The graph of the derivative of a function on $ - 8 \le x \le 5$. Starting well below the x-axis at the left the graph rises steeply, crosses the x-axis near $x = - 7$, reaches a peak at approximately $\left( { - 6,1.5} \right)$, then decreases and just touches the x-axis at $x = - 2$ without crossing it. From there it increases to a peak at approximately $\left( {3,3.5} \right)$ and then falls steeply, crossing the x-axis at $x = 5$ and continuing down off the bottom of the graph.$

Yechim

Biz ehtiyot bo'lishimiz va birinchi ikkita mashq muammosini bajarganimizdek, bu muammoni qilmasligimiz kerak. Berilgan grafik ning grafigi hosila va funksiyaning grafigi emas. Demak, javob faqat grafik qayerda ortib yoki kamayayotganida emas.

Buning o'rniga biz eslashimiz kerakki, hosila belgisi funktsiya qayerda ortib, kamayib borayotganini ko'rsatadi. Agar hosila ijobiy bo'lsa (i.e. uning grafigi $x$ o'qidan yuqori bo'lsa), u holda funktsiya ortib boradi va hosila manfiy bo'lsa (i.e. uning grafigi $x$ o'qi ostida bo'lsa), u holda funktsiya kamayib boradi.

Shunday qilib, grafik $x$ o'qidan yuqorida/pastda qayerda ekanligi juda aniq va shuning uchun bizda quyidagi o'sish/kamayish oraliqlari mavjud.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{{\mbox{Increasing : }}\left( { - 7, - 2} \right)\,\,\,\,\& \,\,\,\left( { - 2,5} \right)\hspace{0.25in}\hspace{0.25in}{\mbox{Decreasing : }}\,\,\left( { - \infty , - 7} \right)\,\,\,\,\,\,\& \,\,\,\,\,\left( {5,\infty } \right)}}\]