Find the Critical Points y=x^2e^(-x)

Solution:

Step:1

Find the first derivative.

Step:1.1

Calculate the derivative of the function.

Step:1.1.1

Apply the Product Rule: $\frac{d}{dx}(f(x)g(x))=f(x)g^{\prime }(x)+g(x)f^{\prime }(x)$, where $f(x)=x^2$ and $g(x)=e^{-x}$.

Step:1.1.2

Use the Chain Rule: $\frac{d}{dx}(f(g(x)))=f^{\prime }(g(x))g^{\prime }(x)$, where $f(u)=e^u$ and $g(x)=-x$.

Step:1.1.2.1

Let $u=-x$. Compute $\frac{d}{du}(e^u)\frac{d}{dx}(-x)$.

Step:1.1.2.2

Apply the Exponential Rule: $\frac{d}{du}(a^u)=a^u\ln (a)$, where $a=e$.

Step:1.1.2.3

Replace $u$ with $-x$ to get $x^2(e^{-x}\frac{d}{dx}(-x))+e^{-x}\frac{d}{dx}(x^2)$.

Step:1.1.3

Differentiate the terms.

Step:1.1.3.1

The derivative of $-x$ is $-1$ since it's a linear term.

Step:1.1.3.2

Apply the Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$, where $n=1$.

Step:1.1.3.3

Simplify the derivative expression.

Step:1.1.3.3.1

Multiply $-1$ by $1$.

Step:1.1.3.3.2

Rearrange the terms to get $-e^{-x}{x}^2+e^{-x}2x$.

Step:1.1.3.3.3

Rewrite $-1e^{-x}$ as $-e^{-x}$.

Step:1.1.3.4

Apply the Power Rule again for $n=2$.

Step:1.1.4

Combine terms to get $f^{\prime }(x)=-x^2{e}^{-x}+2x{e}^{-x}$.

Step:1.2

The derivative $f^{\prime }(x)$ is $-x^2{e}^{-x}+2x{e}^{-x}$.

Step:2

Solve the equation $f^{\prime }(x)=0$.

Step:2.1

Set the derivative equal to zero.

Step:2.2

Factor out the common term $x{e}^{-x}$.

Step:2.2.1

Factor $x{e}^{-x}$ from $-x^2{e}^{-x}$.

Step:2.2.2

Factor $x{e}^{-x}$ from $2x{e}^{-x}$.

Step:2.2.3

Factor out $x{e}^{-x}$ from the entire expression.

Step:2.3

Determine the values that make each factor zero.

Step:2.4

Solve for $x$ in each factor.

Step:2.5

There is no solution for $e^{-x}=0$ as the exponential function never equals zero.

Step:2.6

Solve $-x+2=0$ for $x$.

Step:2.7

The critical points are $x=0$ and $x=2$.

Step:3

There are no points where the derivative is undefined.

Step:4

Evaluate the original function at the critical points.

Step:4.1

Calculate the function value at $x=0$.

Step:4.2

Calculate the function value at $x=2$.

Step:4.3

List all critical points: $(0,0)$ and $(2,\frac{4}{e^2})$.

Knowledge Notes:

1. Product Rule: When differentiating the product of two functions $f(x)$ and $g(x)$, the derivative is $f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$.

2. Chain Rule: Used to differentiate a composite function. If $y=f(g(x))$, then the derivative is $f^{\prime }(g(x))g^{\prime }(x)$.

3. Exponential Rule: The derivative of $a^u$, where $a$ is a constant and $u$ is a function of $x$, is $a^u\ln (a)\cdot u^{\prime }$.

4. Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $n{x}^{n-1}$.

5. Critical Points: Points on the graph of a function where the derivative is zero or undefined. These points are potential locations for local maxima, minima, or points of inflection.

6. Exponential Functions: Functions of the form $a^x$ where $a$ is a positive constant. The exponential function $e^x$ is particularly important due to its unique properties in calculus.

7. Natural Logarithm: The function $\ln (x)$ is the inverse of the exponential function $e^x$. It is undefined for $x\le 0$.

8. Factoring: The process of expressing an expression as a product of its factors. This is useful for solving equations set to zero, as it allows us to apply the Zero Product Property.

9. Zero Product Property: If a product of factors equals zero, at least one of the factors must be zero. This property is used to find the roots of polynomial equations.