Find the Critical Points y=x^3-7x^2-5x+8

Solution:

Step:1

Determine the first derivative of the function.

Step:1.1

Calculate the derivative of the function.

Step:1.1.1

Perform differentiation.

Step:1.1.1.1

Utilize the Sum Rule for differentiation, which implies the derivative of $x^{3} - 7x^{2} - 5x + 8$ is the sum of the derivatives of each term: $\frac{d}{dx}[x^{3}] + \frac{d}{dx}[-7x^{2}] + \frac{d}{dx}[-5x] + \frac{d}{dx}[8]$.

Step:1.1.1.2

Apply the Power Rule, stating that the derivative of $x^{n}$ is $nx^{n-1}$, where $n = 3$. Thus, the derivative becomes $3x^{2} + \frac{d}{dx}[-7x^{2}] + \frac{d}{dx}[-5x] + \frac{d}{dx}[8]$.

Step:1.1.2

Compute the derivative of $-7x^{2}$.

Step:1.1.2.1

Recognize that $-7$ is a constant and use the constant multiple rule to find the derivative, which is $-7\frac{d}{dx}[x^{2}]$.

Step:1.1.2.2

Differentiate $x^{2}$ using the Power Rule where $n = 2$, resulting in $3x^{2} - 7(2x) + \frac{d}{dx}[-5x] + \frac{d}{dx}[8]$.

Step:1.1.2.3

Multiply $2$ by $-7$, simplifying the expression to $3x^{2} - 14x + \frac{d}{dx}[-5x] + \frac{d}{dx}[8]$.

Step:1.1.3

Compute the derivative of $-5x$.

Step:1.1.3.1

Since $-5$ is a constant, apply the constant multiple rule to find the derivative, which is $-5\frac{d}{dx}[x]$.

Step:1.1.3.2

Differentiate $x$ using the Power Rule where $n = 1$, leading to $3x^{2} - 14x - 5 + \frac{d}{dx}[8]$.

Step:1.1.3.3

Multiply $-5$ by $1$, simplifying the expression further to $3x^{2} - 14x - 5 + \frac{d}{dx}[8]$.

Step:1.1.4

Differentiate the constant term using the Constant Rule.

Step:1.1.4.1

The derivative of a constant like $8$ is $0$.

Step:1.1.4.2

Combine the terms to obtain the first derivative: $f'(x) = 3x^{2} - 14x - 5$.

Step:1.2

The first derivative of $f(x)$ is $f'(x) = 3x^{2} - 14x - 5$.

Step:2

Solve for $x$ when the first derivative is zero: $3x^{2} - 14x - 5 = 0$.

Step:2.1

Set the first derivative to zero.

Step:2.2

Factor the quadratic equation.

Step:2.2.1

For a quadratic $ax^{2} + bx + c$, find two numbers that multiply to $ac$ and add to $b$. Here, $ac = 3(-5) = -15$ and $b = -14$.

Step:2.2.1.1

Factor out $-14x$.

Step:2.2.1.2

Decompose $-14x$ into two terms that add up to $-14x$ and multiply to $-15$.

Step:2.2.1.3

Apply the distributive property to split the middle term.

Step:2.2.1.4

Multiply $x$ by $1$.

Step:2.2.2

Group terms and factor out the greatest common factor from each group.

Step:2.2.2.1

Group the terms into two pairs.

Step:2.2.2.2

Factor the greatest common factor from each pair.

Step:2.2.3

Factor the expression by extracting the common factor $3x + 1$.

Step:2.3

Set each factor equal to zero.

Step:2.4

Solve $3x + 1 = 0$ for $x$.

Step:2.4.1

Set the first factor equal to zero.

Step:2.4.2

Isolate $x$.

Step:2.4.2.1

Subtract $1$ from both sides.

Step:2.4.2.2

Divide by $3$.

Step:2.4.2.2.1

Divide each term by $3$.

Step:2.4.2.2.2

Simplify the equation.

Step:2.4.2.2.2.1

Cancel common factors.

Step:2.4.2.2.2.1.1

Reduce the fraction.

Step:2.4.2.2.2.1.2

Solve for $x$.

Step:2.4.2.2.3

Simplify the right side.

Step:2.5

Solve $x - 5 = 0$ for $x$.

Step:2.5.1

Set the second factor equal to zero.

Step:2.5.2

Add $5$ to both sides.

Step:2.6

The solutions are $x = -\frac{1}{3}$ and $x = 5$.

Step:3

Check for points where the derivative does not exist. There are none in this case.

Step:4

Evaluate the original function at the critical points.

Step:4.1

Evaluate at $x = -\frac{1}{3}$.

Step:4.1.1

Substitute $x$ with $-\frac{1}{3}$ in the original function.

Step:4.1.2

Simplify the expression.

Step:4.2

Evaluate at $x = 5$.

Step:4.2.1

Substitute $x$ with $5$ in the original function.

Step:4.2.2

Simplify the expression.

Step:4.3

Compile all critical points: $\left(-\frac{1}{3}, \frac{239}{27}\right)$ and $(5, -67)$.

Step:5

The critical points of the function $y = x^3 - 7x^2 - 5x + 8$ are at $x = -\frac{1}{3}$ and $x = 5$.

Knowledge Notes:

1. Derivative: The derivative of a function at a point is the rate at which the function's value changes at that point. It is a fundamental tool in calculus for analyzing the behavior of functions.

2. Sum Rule: This rule states that the derivative of a sum of functions is the sum of the derivatives of those functions.

3. Power Rule: A basic differentiation rule that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

4. Constant Multiple Rule: If $c$ is a constant and $f(x)$ is a function, then the derivative of $cf(x)$ is $cf'(x)$.

5. Constant Rule: The derivative of a constant is zero.

6. Factoring: A method used to simplify expressions and solve equations by expressing a polynomial as the product of its factors.

7. Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, can be solved by factoring, completing the square, or using the quadratic formula.

8. Critical Points: Points on a graph where the derivative is zero or undefined. These points can indicate local maxima, minima, or points of inflection.