Determine without graphing, whether the given function has a maximum value and then find the value. f(x)=3x^2+30x-1
Steve had already answered this question for you.
http://www.jiskha.com/display.cgi?id=1363360907
What part of his answer did you not like ?
I needed help learning how to find the value.
f(x) = 3(x^2 + 10x +25)-76
= 3(x+5)^2 -76
The function has no maximum. It has a minimum of -76 when x = -5.
Hmmm. I explained that there is no maximum value. Don't expect to be able to find it.
Now, if you want to find the minimum value, try completing the square:
3x^2+30x-1
= 3(x^2 + 10x) - 1
= 3(x^2 + 10x + 25) - 1 - 75
That step is the key. We added 75 to complete the square, so we have to subtract it as well to avoid changing f(x)
= 3(x+5)^2 - 76
Now you can see that since (x+5)^2 is always at least zero (since squares are never negative), the smallest value for (x+5)^2 is when x = -5. In that case, f(x) = 3(0) - 76
So, f(-5) = -76, and it can never be less than that, because whatever x is, (x+5)^2 will be some positive number, making f(x) > -76
-80, 5
To determine whether the given function, f(x) = 3x^2 + 30x - 1, has a maximum value, we can analyze the coefficient of the x^2 term.
The coefficient of x^2 is positive (3), which means the graph of the function opens upwards, forming a "U" shape. In this case, the function will have a minimum value, not a maximum.
To find the minimum value, we can use the vertex form of a quadratic function, which is given by f(x) = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.
In our given function, f(x) = 3x^2 + 30x - 1, we can first factor out the common factor of 3 from the x^2 and x terms, which gives us:
f(x) = 3(x^2 + 10x) - 1
Next, we complete the square inside the parentheses. We take half of the coefficient of the x term (10/2 = 5), square it (5^2 = 25), and add it inside the parentheses. However, since we added 25 inside the parentheses, we also need to subtract 3 * 25 = 75 from the expression outside the parentheses to maintain the equation's balance.
f(x) = 3(x^2 + 10x + 25 - 25) - 1
= 3((x + 5)^2 - 25) - 1
= 3(x + 5)^2 - 75 - 1
= 3(x + 5)^2 - 76
Now we have the function in the vertex form: f(x) = 3(x + 5)^2 - 76. Comparing this form with the general form (f(x) = a(x-h)^2 + k), we can see that the vertex of the parabola is located at the point (-5, -76). Therefore, the minimum value of the function f(x) = 3x^2 + 30x - 1 is -76.
So, the function f(x) = 3x^2 + 30x - 1 has a minimum value of -76.