Find the Local Maxima and Minima h(x)=3x^2+10x+3

$h (x) = 3 x^{2} + 10 x + 3$

Step 1

Find the first derivative of the function.

By the Sum Rule, the derivative of $3 x^{2} + 10 x + 3$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 (2 x) + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$

Evaluate $\frac{d}{d x} [10 x]$ .

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$6 x + 10 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

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

$6 x + 10 \cdot 1 + \frac{d}{d x} [3]$

Multiply $10$ by $1$ .

$6 x + 10 + \frac{d}{d x} [3]$

Differentiate using the Constant Rule.

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$6 x + 10 + 0$

Add $6 x + 10$ and $0$ .

$6 x + 10$

Step 2

Find the second derivative of the function.

By the Sum Rule, the derivative of $6 x + 10$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [10]$ .

$h'' (x) = \frac{d}{d x} (6 x) + \frac{d}{d x} (10)$

Evaluate $\frac{d}{d x} [6 x]$ .

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$h'' (x) = 6 \frac{d}{d x} (x) + \frac{d}{d x} (10)$

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

$h'' (x) = 6 \cdot 1 + \frac{d}{d x} (10)$

Multiply $6$ by $1$ .

$h'' (x) = 6 + \frac{d}{d x} (10)$

Differentiate using the Constant Rule.

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$h'' (x) = 6 + 0$

Add $6$ and $0$ .

$h'' (x) = 6$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$6 x + 10 = 0$

Step 4

Find the first derivative.

By the Sum Rule, the derivative of $3 x^{2} + 10 x + 3$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [10 x] + \frac{d}{d x} [3]$ .

$f' (x) = \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (10 x) + \frac{d}{d x} (3)$

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (10 x) + \frac{d}{d x} (3)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = 3 (2 x) + \frac{d}{d x} (10 x) + \frac{d}{d x} (3)$

Multiply $2$ by $3$ .

$f' (x) = 6 x + \frac{d}{d x} (10 x) + \frac{d}{d x} (3)$

Evaluate $\frac{d}{d x} [10 x]$ .

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$f' (x) = 6 x + 10 \frac{d}{d x} (x) + \frac{d}{d x} (3)$

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

$f' (x) = 6 x + 10 \cdot 1 + \frac{d}{d x} (3)$

Multiply $10$ by $1$ .

$f' (x) = 6 x + 10 + \frac{d}{d x} (3)$

Differentiate using the Constant Rule.

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f' (x) = 6 x + 10 + 0$

Add $6 x + 10$ and $0$ .

$f' (x) = 6 x + 10$

The first derivative of $h (x)$ with respect to $x$ is $6 x + 10$ .

$6 x + 10$

Step 5

Set the first derivative equal to $0$ then solve the equation $6 x + 10 = 0$ .

Set the first derivative equal to $0$ .

$6 x + 10 = 0$

Subtract $10$ from both sides of the equation.

$6 x = - 10$

Divide each term in $6 x = - 10$ by $6$ and simplify.

Divide each term in $6 x = - 10$ by $6$ .

$\frac{6 x}{6} = \frac{- 10}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 10}{6}$

Divide $x$ by $1$ .

$x = \frac{- 10}{6}$

Simplify the right side.

Cancel the common factor of $- 10$ and $6$ .

Factor $2$ out of $- 10$ .

$x = \frac{2 (- 5)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = \frac{2 \cdot - 5}{2 \cdot 3}$

Cancel the common factor.

$x = \frac{2 \cdot - 5}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{- 5}{3}$

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 6

Find the values where the derivative is undefined.

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 7

Critical points to evaluate.

$x = - \frac{5}{3}$

Step 8

Evaluate the second derivative at $x = - \frac{5}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6$

Step 9

$x = - \frac{5}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{5}{3}$ is a local minimum

Step 10

Find the y-value when $x = - \frac{5}{3}$ .

Replace the variable $x$ with $- \frac{5}{3}$ in the expression.

$h (- \frac{5}{3}) = 3 {(- \frac{5}{3})}^{2} + 10 (- \frac{5}{3}) + 3$

Simplify the result.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{5}{3}$ .

$h (- \frac{5}{3}) = 3 ({(- 1)}^{2} {(\frac{5}{3})}^{2}) + 10 (- \frac{5}{3}) + 3$

Apply the product rule to $\frac{5}{3}$ .

$h (- \frac{5}{3}) = 3 ({(- 1)}^{2} (\frac{5^{2}}{3^{2}})) + 10 (- \frac{5}{3}) + 3$

Raise $- 1$ to the power of $2$ .

$h (- \frac{5}{3}) = 3 (1 (\frac{5^{2}}{3^{2}})) + 10 (- \frac{5}{3}) + 3$

Multiply $\frac{5^{2}}{3^{2}}$ by $1$ .

$h (- \frac{5}{3}) = 3 (\frac{5^{2}}{3^{2}}) + 10 (- \frac{5}{3}) + 3$

Raise $5$ to the power of $2$ .

$h (- \frac{5}{3}) = 3 (\frac{25}{3^{2}}) + 10 (- \frac{5}{3}) + 3$

Raise $3$ to the power of $2$ .

$h (- \frac{5}{3}) = 3 (\frac{25}{9}) + 10 (- \frac{5}{3}) + 3$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$h (- \frac{5}{3}) = 3 (\frac{25}{3 (3)}) + 10 (- \frac{5}{3}) + 3$

Cancel the common factor.

$h (- \frac{5}{3}) = 3 (\frac{25}{3 \cdot 3}) + 10 (- \frac{5}{3}) + 3$

Rewrite the expression.

$h (- \frac{5}{3}) = \frac{25}{3} + 10 (- \frac{5}{3}) + 3$

Multiply $10 (- \frac{5}{3})$ .

Multiply $- 1$ by $10$ .

$h (- \frac{5}{3}) = \frac{25}{3} - 10 (\frac{5}{3}) + 3$

Combine $- 10$ and $\frac{5}{3}$ .

$h (- \frac{5}{3}) = \frac{25}{3} + \frac{- 10 \cdot 5}{3} + 3$

Multiply $- 10$ by $5$ .

$h (- \frac{5}{3}) = \frac{25}{3} + \frac{- 50}{3} + 3$

Move the negative in front of the fraction.

$h (- \frac{5}{3}) = \frac{25}{3} - \frac{50}{3} + 3$

Combine fractions.

Combine the numerators over the common denominator.

$h (- \frac{5}{3}) = 3 + \frac{25 - 50}{3}$

Simplify the expression.

Subtract $50$ from $25$ .

$h (- \frac{5}{3}) = 3 + \frac{- 25}{3}$

Move the negative in front of the fraction.

$h (- \frac{5}{3}) = 3 - \frac{25}{3}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$h (- \frac{5}{3}) = 3 \cdot \frac{3}{3} - \frac{25}{3}$

Combine $3$ and $\frac{3}{3}$ .

$h (- \frac{5}{3}) = \frac{3 \cdot 3}{3} - \frac{25}{3}$

Combine the numerators over the common denominator.

$h (- \frac{5}{3}) = \frac{3 \cdot 3 - 25}{3}$

Simplify the numerator.

Multiply $3$ by $3$ .

$h (- \frac{5}{3}) = \frac{9 - 25}{3}$

Subtract $25$ from $9$ .

$h (- \frac{5}{3}) = \frac{- 16}{3}$

Move the negative in front of the fraction.

$h (- \frac{5}{3}) = - \frac{16}{3}$

The final answer is $- \frac{16}{3}$ .

$y = - \frac{16}{3}$

Step 11

These are the local extrema for $h (x) = 3 x^{2} + 10 x + 3$ .

$(- \frac{5}{3}, - \frac{16}{3})$ is a local minima

Step 12

Find the Local Maxima and Minima h(x)=3x^2+10x+3

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