Calculus Examples

$f (x) = x^{a} {(1 - x)}^{b}$ , $0 \leq x \leq 1$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{a}$ and $g (x) = {(1 - x)}^{b}$ .

$x^{a} \frac{d}{d x} [{(1 - x)}^{b}] + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{b}$ and $g (x) = 1 - x$ .

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Step 1.1.1.2.1

To apply the Chain Rule, set $u$ as $1 - x$ .

$x^{a} (\frac{d}{d u} [u^{b}] \frac{d}{d x} [1 - x]) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = b$ .

$x^{a} (b u^{b - 1} \frac{d}{d x} [1 - x]) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.2.3

Replace all occurrences of $u$ with $1 - x$ .

$x^{a} (b {(1 - x)}^{b - 1} \frac{d}{d x} [1 - x]) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.3

Differentiate.

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Step 1.1.1.3.1

By the Sum Rule, the derivative of $1 - x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x]$ .

$x^{a} (b {(1 - x)}^{b - 1} (\frac{d}{d x} [1] + \frac{d}{d x} [- x])) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$x^{a} (b {(1 - x)}^{b - 1} (0 + \frac{d}{d x} [- x])) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.3.3

Add $0$ and $\frac{d}{d x} [- x]$ .

$x^{a} (b {(1 - x)}^{b - 1} \frac{d}{d x} [- x]) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$x^{a} (b {(1 - x)}^{b - 1} (- \frac{d}{d x} [x])) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{a} (b {(1 - x)}^{b - 1} (- 1 \cdot 1)) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.3.6

Simplify the expression.

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Step 1.1.1.3.6.1

Multiply $- 1$ by $1$ .

$x^{a} (b {(1 - x)}^{b - 1} \cdot - 1) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.3.6.2

Move $- 1$ to the left of $b {(1 - x)}^{b - 1}$ .

$x^{a} (- 1 \cdot (b {(1 - x)}^{b - 1})) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.3.6.3

Rewrite $- 1 b$ as $- b$ .

$x^{a} (- b {(1 - x)}^{b - 1}) + {(1 - x)}^{b} \frac{d}{d x} [x^{a}]$

Step 1.1.1.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = a$ .

$x^{a} (- b {(1 - x)}^{b - 1}) + {(1 - x)}^{b} (a x^{a - 1})$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

Reorder terms.

$- x^{a} b {(1 - x)}^{b - 1} + x^{a - 1} a {(1 - x)}^{b}$

Step 1.1.1.4.2

Reorder factors in $- x^{a} b {(1 - x)}^{b - 1} + x^{a - 1} a {(1 - x)}^{b}$ .

$f' (x) = - b x^{a} {(1 - x)}^{b - 1} + a x^{a - 1} {(1 - x)}^{b}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- b x^{a} {(1 - x)}^{b - 1} + a x^{a - 1} {(1 - x)}^{b}$ .

$- b x^{a} {(1 - x)}^{b - 1} + a x^{a - 1} {(1 - x)}^{b}$

Step 1.2

Set the first derivative equal to $0$ .

$- b x^{a} {(1 - x)}^{b - 1} + a x^{a - 1} {(1 - x)}^{b} = 0$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = 0$ .

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Step 2.1.1

Substitute $0$ for $x$ .

${(0)}^{a} {(1 - (0))}^{b}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Subtract $0$ from $1$ .

$0^{a} \cdot 1^{b}$

Step 2.1.2.2

One to any power is one.

$0^{a} \cdot 1$

Step 2.1.2.3

Multiply $0^{a}$ by $1$ .

$0^{a}$

Step 2.2

Evaluate at $x = 1$ .

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Step 2.2.1

Substitute $1$ for $x$ .

${(1)}^{a} {(1 - (1))}^{b}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

One to any power is one.

$1 {(1 - (1))}^{b}$

Step 2.2.2.2

Multiply ${(1 - (1))}^{b}$ by $1$ .

${(1 - (1))}^{b}$

Step 2.2.2.3

Multiply $- 1$ by $1$ .

${(1 - 1)}^{b}$

Step 2.2.2.4

Subtract $1$ from $1$ .

$0^{b}$

Step 2.3

List all of the points.

$(0, 0^{a}), (1, 0^{b})$

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

No absolute maximum

No absolute minimum

Step 4