國際數學圖譜

IAL 2022 Jan Q1

A Level / Edexcel / P3

IAL 2022 Jan Paper · Question 1

题目

Problem

Find, using calculus, the $x$ coordinate of the stationary point on the curve with equation

y=(2x+5)e^{3x}

(4)

题目中文翻译

用微积分求曲线

y=(2x+5)e^{3x}

的驻点的 $x$ 坐标。

解答

We have

y=(2x+5)e^{3x}

Using the product rule,

\frac{dy}{dx} =2e^{3x}+(2x+5)\cdot 3e^{3x}

So

\frac{dy}{dx} =e^{3x}\left[2+3(2x+5)\right]

Simplifying,

\frac{dy}{dx} =e^{3x}(6x+17)

At a stationary point,

\frac{dy}{dx}=0

Thus

e^{3x}(6x+17)=0

Since

e^{3x}>0

we must have

6x+17=0

Therefore

x=-\frac{17}{6}

So the $x$ coordinate of the stationary point is

\boxed{-\frac{17}{6}}