Which defines a circle

Find the local maximum and minimum values of f using both the first and second derivative tests. f(x) = 6 9x2 − 6x3 local maximu

olga55 [171]

The local minimum value of the function f(x) = 6 + 9x² - 6x³ is 6 and the local maximum value of the function f(x) = 6 + 9x² - 6x³ is 9.

For given question,

We have been given a function f(x) = 6 + 9x² - 6x³

We need to find the local maximum and local minimum of the function f(x)

First we find the first derivative of the function.

⇒ f'(x) = 0 + 18x - 18x²

⇒ f'(x) = - 18x² + 18x

Putting the first derivative of the function equal to zero, we get

⇒ f'(x) = 0

⇒ - 18x² + 18x = 0

⇒ 18(-x² + x) = 0

⇒ x (-x + 1) = 0

⇒ x = 0 or -x + 1 = 0

⇒ x = 0 or x = 1

Now we find the second derivative of the function.

⇒ f"(x) = - 36x + 18

At x = 0 the value of second derivative of function f(x),

⇒ f"(0) = - 36(0) + 18

⇒ f"(0) = 0 + 18

⇒ f"(0) = 18

Here, at x=0, f"(x) > 0

This means, the function f(x) has the local minimum value at x = 0, which is given by

⇒ f(0) = 6 + 9(0)² - 6(0)³

⇒ f(0) = 6 + 0 - 0

⇒ f(0) = 6

At x = 1 the value of second derivative of function f(x),

⇒ f"(1) = - 36(1) + 18

⇒ f"(1) = - 18

Here, at x = 1, f"(x) < 0

This means, the function f(x) has the local maximum value at x = 1, which is given by

⇒ f(1) = 6 + 9(1)² - 6(1)³

⇒ f(1) = 6 + 9 - 6

⇒ f(1) = 9

So, the function f(x) = 6 + 9x² - 6x³ has local minimum at x = 0 and local maximum at x = 1.

Therefore, the local minimum value of the function f(x) = 6 + 9x² - 6x³ is 6 and the local maximum value of the function f(x) = 6 + 9x² - 6x³ is 9.

Learn more about the local minimum value and local maximum value here:

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3 0

2 years ago

Find the value of x and the measure of each angle in the parallelogram

Morgarella [4.7K]

Answer:

$\angle D= 162\\\\\angle E = 18\\\\\angle F = 162\\\\\angle G = 18$

Step-by-step explanation:

By properties of parallelograms we know that opposite are the same so we have

$12x=4(x+3)\\\\4(3x)=4(x+3)\\\\3x=x+3\\\\2x=3\\\\x=\frac{3}{2}$

so for the angles we have

$\angle G=\angle E=12(\frac{3}{2})=18\\\\\angle D=\angle F = 180-\angle G=180-18=162$

5 0

3 years ago

Apply the distributive property to factor out the greatest common factor. 6+30=6+30=6, plus, 30, equals

kodGreya [7K]

Step-by-step explanation:

the distributive property is 6(1+5), the greatest common factor is 6

8 0

3 years ago

Find the 78th term of the arithmetic sequence -4, -2, 0, ...−4,−2,0,...

Tasya [4]

Answer:

0

Step-by-step explanation:

-4, -2, 0

the 3rd number is always a 0 so if you do the sequence 78 times the 78th term will be a 0

7 0

2 years ago

What is the product of the polynomial 2x^3 + 8x^2 + x + 3 times the<br> monomial x?

Elis [28]

Answer:

$\huge\boxed{2x^4 + 8x^3 + x^2 + 3x}$

Step-by-step explanation:

When we multiply a polynomial by a specific monomial, all terms in the polynomial get multiplied by the monomial.

So we can do this by term.

$2x^3 \cdot x$

Multiplying x by x to the something power will just increase the exponent on the x term by one (since powers are just x multiplied by itself). This means the exponent here will increase by 1, so this becomes $2x^4$ .

Same logic applies to the second term: $8x^2$ , raise the exponent by one, $8x^3$ .

For the third term, if there is no exponent on a term you can assume it's exponent is 1 (so $x^1$ in this case - as $x^1=x$ ). Again, using the same logic, this turns into $x^2$ .

For the last term, when we multiply a constant by a variable, we get a term with a variable and a coefficient - the constant becomes the coefficient and the variable stays the variable. Therefore, $3 \cdot x = 3x$ .

Adding these terms all together gets us $2x^4 + 8x^3 + x^2 + 3x$ .

Hope this helped!

8 0

4 years ago

Which defines a circle​

Which defines a circle