Question

For what value of x will the function f(x) = -3(x - 10)(x - 4) have a maximum value? Find the maximum value.

Answer 1

Hello!

To find the maximum value of the function f(x) = -3(x - 10)(x - 4), the easiest way is to find the vertex using the formula: x = -b/2a.

Firstly, we need to simplify f(x).

f(x) = -3(x - 10)(x - 4)

f(x) = -3(x² - 14x + 40)

f(x) = -3x² + 42x + -120

Since the equation f(x) is now simplified to standard form, we can find the vertex.

a = -3, b = 42, and c = -120

x = -(42)/2(-3) = -42/-6 = 7

Then, we substitute 7 into the the function f(x) = -3(x - 10)(x - 4), or

f(x) = -3x² + 42x + -120, to find the y-value of the vertex.

f(x) = -3(7 - 10)(7 - 4)

f(x) = -3(-3)(4)

f(x) = 27

The vertex of f(x) is (7, 27).

Therefore, the maximum x-value for the function f(x) is 7.

Answer 2

Maximum point occurs at the line of symmetry

Step 1: Find the x-intercept. [ x- intercept = when f(x) = 0 ]

f(x) = -3(x - 10)(x - 4)

-3(x - 10)(x - 4) = 0

x - 10 = 0 or x - 4 = 0

x = 10 or x = 4

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Step 2 : Find the line of symmetry [ Midpoint of x ]

midpoint of x = (10 + 4) ÷ 2 = 7

Line of symmetry : x = 7

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Answer: Maximum value occurs when x = 7