A function $f$ has domain $ ext{R}$ and range $ ext{y} ext{∈} ext{R: y} ext{≥ e}.$ The graph of $y = f(x)$ is shown. The gradient of the curve at the point $(x, y)$ is given by $rac{dy}{dx} = (x-1)e^x$. Find an expression for $f(x)$. Fully justify your answer.

A-Level AQA Maths Pure Implicit Differentiation: A function $f$ has domain $ ext{

A function $f$ has domain $ ext{R}$ and range $ ext{y} ext{∈} ext{R: y} ext{≥ e}.$ The graph of $y = f(x)$ is shown - AQA - A-Level Maths Pure - Question 7 - 2018 - Paper 2

Question 7

A function $f$ has domain $ ext{R}$ and range $ ext{y} ext{∈} ext{R: y} ext{≥ e}.$ The graph of $y = f(x)$ is shown. The gradient of the curve at the point $(x,... show full transcript

Worked Solution & Example Answer:A function $f$ has domain $ ext{R}$ and range $ ext{y} ext{∈} ext{R: y} ext{≥ e}.$ The graph of $y = f(x)$ is shown - AQA - A-Level Maths Pure - Question 7 - 2018 - Paper 2

Step 1

Integrate the gradient function

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Answer

To find the expression for $f(x)$ , we start by integrating the gradient function:

$\frac{dy}{dx} = (x-1)e^x$

Using integration by parts, let:

$u = (x-1)$ , then $du = dx$ ,
$dv = e^x dx$ , then $v = e^x$ .

Applying integration by parts: $\int u \, dv = uv - \int v \, du$

Substituting in, $\int (x-1)e^x \, dx = (x-1)e^x - \int e^x \, dx$

= $(x-1)e^x - e^x + C$ .

Thus, we have: $f(x) = (x-2)e^x + C$

Step 2

Identify the minimum value of y

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Answer

To find the minimum value of $y$ , we look at the condition where the gradient $rac{dy}{dx} = 0$ :

$(x-1)e^x = 0$

This occurs when $x = 1.$

At this point, substitute $x = 1$ into $f(x)$ :

$f(1) = (1-2)e^1 + C = -e + C$

From the range of the function given, we know that the minimum value of $y$ must equal $e$ at the minimum point. Therefore:

$-e + C = e ightarrow C = 2e.$

Step 3

State the final expression for f(x)

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Answer

Combining the results, we get:

$f(x) = (x-2)e^x + 2e$

This represents the function $f(x)$ that satisfies the given conditions.

A function $f$ has domain $ ext{R}$ and range $ ext{y} ext{∈} ext{R: y} ext{≥ e}.$ The graph of $y = f(x)$ is shown - AQA - A-Level Maths Pure - Question 7 - 2018 - Paper 2

Worked Solution & Example Answer:A function $f$ has domain $ ext{R}$ and range $ ext{y} ext{∈} ext{R: y} ext{≥ e}.$ The graph of $y = f(x)$ is shown - AQA - A-Level Maths Pure - Question 7 - 2018 - Paper 2

Integrate the gradient function

Identify the minimum value of y

State the final expression for f(x)

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