Question

Given the following function, (a) find the vertex; (b) determine whether there is a maximum or a minimum value, and find the value; (c) find the range, and (d) find the intervals on which the function is increasing and the intervals on which the function is decreasing. f(x) = - 4x2 - 20x + 7

Answer 1

Solution-

Consider the function

f(x) = -4x² -20x + 7 .....(1)

(a)

At vertex differentiation of f(x) is equal to 0.

So, f'(x) = 0 implies

(d/dx)(-4x² -20x +7) =0

Or -4(2x) -20(1) + 7(0) = 0

Or -8x -20 +0 =0 .....(2)

Or -8x = 20

Or x = 20/(-8)

Or x = -2.5

Putting x = -2.5 in equation (1) to get

f(-2.5) = -4(-2.5)² -20(-2.5) + 7

f(-2.5) = -4(6.25) + 50 +7

f(-2.5) = -25 + 50 +7 = 27

So, Vertex of the function is (-2.5, 27).

Hence, Vertex = (-2.5, 27)

(b)

To determine whether it is maximum or minimum point, let us double differentiate the function f(x) =

Since from equation (2)

f'(x) = -8x + 20

Differentiating it again to get

f"(x) = (d/dx)(-8x + 20)

f"(x) = -8(1) + 0

f"(x) = -8

Since double differentiation is always negative (-8).

So, x = -2.5 is the point of maximum value.

And the maximum value is f(-2.5) = 27.

Hence , x = -2.5 is the point of maximum value with value = f(-2.5) = 27 .

(c)

Since f(x) = 27 is the maximum value of y and there is no minimum value of f(x).

So, range of function is y belongs to (-∞, 27).

Hence, Range = (-∞, 27) .

(d)

Since x = -2.5 and f(x) = 27 is the only point of optimum (maximum) value.

So both sides of x = -2.5 the graph is either increasing or decreasing.

So, let us check

At x = 0

f(x) = -4(0)² -20(0) + 7 = 7

And at x = 5

f(x) = -4(5)² -20(5) +7 = -100 -100 +7 = 7

So, we can say that from x = -∞ to -2.5 the function is increasing (value of f(x) is increasing with x) and from -2.5 to ∞ , the function is decreasing (value of f(x) is decresing with x) .

Hence,

Function is increasing on (-∞, -2.5) . And

Function is decreasing on ( -2.5 , ∞) .