Consider the function 𝑓(𝑥)=2𝑥2+4𝑥−1. Which of the following statements is FALSE?

Question

Select one:

This function is concave up everywhere

This function is increasing at x=1

This function has a horizontal tangent line at x=0

This function is increasing at a faster rate at x=3 than x=2.

Answer 1

Let f be the function defined on R by:

f(x) = 2x² + 4x - 1

The derivative of f is given on R by:

f'(x) = 4x + 4

f'(0) = 4 * 0 + 4 = 4 ≠ 0

The statement "This function has a horizontal tangent line at x = 0" is FALSE.

(In details, the function has a horizontal tangent line at a given point would mean that the tangent line has slope 0 which would mean that the derivative at that point is 0, which is not the case here).



f'(x) > 0

⇒ 4x + 4 > 0

⇒ 4x + 4 - 4 > 0 - 4

⇒ 4x > -4

⇒ 4x / 4 > -4 / 4

⇒ x > -1

So the function f is increasing for x > -1 and decreasing for x < -1.

The statement "This function is increasing at x = 1" is TRUE.

f'(3) = 4 * 3 + 4 = 12 + 4 = 16

f'(2) = 4 * 2 + 4 = 8 + 4 = 12

f'(3) > f'(2)

Keeping in mind that the derivative at a given point is the instantaneous rate of change at that point,

the statement "This function is increasing at a faster rate at x=3 than x=2" is TRUE.

The second derivative of f is given on R by:

f''(x) = 4

For all x in R, f''(x) > 0.

The statement "This function is concave up everywhere" is TRUE.