Find the open intervals where the function is concave up and concave down. y=-2x^2+12x-18

Question:

Find the open intervals where the function is concave up and concave down.

{eq}y=-2x^2+12x-18 {/eq}

Concavity:

Imagine a parabola that is either opens upward or opens downward. When the parabola opens upward, the function is said to concave up. Conversely, a parabola facing downward is said to concave down. This definition can be extended to any polynomial order to determine concavity.

Answer and Explanation:


Given the function:

{eq}\displaystyle \rm y = -2x^2 + 12x + 8 {/eq}

We can determine the concavity by using the second derivative test. The test is written out as follows:

  • Concave up: {eq}\displaystyle \rm \frac{d^2 y}{dx^2} > 0 {/eq}
  • Concave down: {eq}\displaystyle \rm \frac{d^2 y}{dx^2} < 0 {/eq}


We now take the second derivative of our function:

{eq}\displaystyle \rm \frac{d^2 y}{dx^2} = \frac{d}{dx} \frac{d}{dx} \Big[ -2x^2 + 12x + 8 \Big] {/eq}

This process essentially means we differentiate our function twice. By recalling the derivative of a polynomial term,

  • {eq}\displaystyle \rm \frac{d}{dx} x^n = nx^{n-1} {/eq}

We can determine the second derivative to be:

{eq}\displaystyle \rm \frac{d^2 y}{dx^2} = -4 {/eq}

Since our second derivative is constant and negative, this means that for all x the function concaves down.


Learn more about this topic:

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Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5
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