What is the minimum value of the function f(x)= 4x^2+24x+39
To find the minimum value of the function f(x) = 4x^2 + 24x + 39, we can use the concept of completing the square or the derivative test.
Method 1: Completing the Square
Step 1: Rewrite the function in the form f(x) = a(x - h)^2 + k. To do this, we need to complete the square.
f(x) = 4x^2 + 24x + 39
= 4(x^2 + 6x) + 39
= 4(x^2 + 6x + 9 - 9) + 39 (Since (a + b)^2 = a^2 + 2ab + b^2, we need to add and subtract 9 to complete the square.)
= 4((x + 3)^2 - 9) + 39
= 4(x + 3)^2 - 36 + 39
= 4(x + 3)^2 + 3
Step 2: Since the term (x + 3)^2 is squared, it is always non-negative, so the minimum value occurs when it equals to zero, which is when x = -3.
Step 3: Substitute x = -3 into the function to find the minimum value.
f(-3) = 4(-3 + 3)^2 + 3
= 4 * 0 + 3
= 3
Therefore, the minimum value of the function f(x) = 4x^2 + 24x + 39 is 3.
Method 2: Derivative Test
Step 1: Take the derivative of the function f(x) to find the critical points.
f(x) = 4x^2 + 24x + 39
f'(x) = 8x + 24
Step 2: Set the derivative equal to zero and solve for x to find the critical point.
8x + 24 = 0
8x = -24
x = -3
Therefore, the critical point is x = -3.
Step 3: Use the derivative test to determine whether the critical point is a minimum.
For this, we need to evaluate the second derivative of the function at the critical point.
f''(x) = 8
Since the second derivative f''(x) = 8 is positive, the function has a minimum at x = -3.
Step 4: Substitute x = -3 into the function f(x) to find the minimum value.
f(-3) = 4(-3)^2 + 24(-3) + 39
= 4 * 9 - 72 + 39
= 36 - 72 + 39
= 3
Therefore, the minimum value of the function f(x) = 4x^2 + 24x + 39 is 3.
this equation is writing as ax^2+bx+c
you need to find x. the equation for x is x=b/2a
plug it in b=24 a=4
x=24/2x4=24/8=3
if x = 3 plug it in to your equation
4(3)^2+24(3)+39
4x9+72+39
36+72+39=147
so the vertex is (3,147)
because a is 24 its positive meaning it opens up makeing the minimum value the vertex(3,147)