How do you find the maximum value of #f(x) = -3x^2 +30x -50#?

Question

Charles Arocha · Accepted Answer

Maximum value of # f(x) # is #25#

#f (x) = -3x^2 +30x -50 #

#f^' (x) = -6x +30 # , At stationary point #f^' (x) = 0# or

#-6x +30 = 0 or x = 30/6 =5 #

At #x=5 ; f(x) = - 3(5^2)+30*(5) -50= -75+150-50=25#

Stationary point is # (5, 25) #

2nd derivative test for maximum or minimum :

# (d^2y)/dx^2 = -6 # (negative)

If #(d^2y)/dx^2# is negative, then the point is a maximum turning point.

So #(5,25) # is maximum turning point and maximum value of

# f(x) # is #25#

graph{-3x^2+30x-50 [-50.64, 50.6, -25.28, 25.37]} [Ans]

Scarlett Allison · Answer

undefined To find the maximum value of $ f(x) = -3x^2 + 30x - 50 $, follow these steps:

1. Identify the coefficient of the quadratic term, $ -3 $, which is negative. This indicates that the parabola opens downwards, meaning it has a maximum value.

2. To find the x-coordinate of the maximum point, use the formula for the x-coordinate of the vertex of a quadratic function: $ x = \frac{-b}{2a} $, where $ a = -3 $ and $ b = 30 $.

3. Substitute the values of $ a $ and $ b $ into the formula: $ x = \frac{-30}{2(-3)} $.

4. Calculate $ x $ to find the x-coordinate of the vertex.

5. Once you have the x-coordinate of the vertex, substitute it back into the original function $ f(x) $ to find the corresponding y-coordinate.

6. The y-coordinate of the vertex gives the maximum value of the function $ f(x) $.

Follow these steps to determine the maximum value of the function $ f(x) = -3x^2 + 30x - 50 $.

Paisley Johnson · Answer

undefined To find the maximum value of $ f(x) = -3x^2 + 30x - 50 $, you can follow these steps:

1. Identify the coefficient of the quadratic term ($ x^2 $), which is -3 in this case.
2. Determine the x-coordinate of the vertex using the formula $ x = \frac{-b}{2a} $, where $ a = -3 $ and $ b = 30 $.
3. Calculate the x-coordinate of the vertex using the formula from step 2.
4. Substitute the x-coordinate of the vertex into the original function to find the maximum value of f(x).