Increasing and decreasing functions questions with answers

Questions

1. Question 1Increasing or decreasing at a point

   Determine whether $y = x^{3} - 6x^{2} + 9x + 1$ is increasing or decreasing at $x = 0$.

   Show the worked solution

   1. Step 1: Differentiate$\dfrac{dy}{dx} = 3x^{2} - 6(2)x + 9(1)$$= 3x^{2} - 12x + 9$
   2. Step 2: Put the given x into dy/dx$\dfrac{dy}{dx} = 3(0)^{2} - 12(0) + 9$$= 9$
   3. Step 3: Increasing or decreasing? Use the sign of dy/dx$\dfrac{dy}{dx} > 0$, so $y$ is increasing at $x = 0$.
2. Question 2Increasing or decreasing at a point

   Determine whether $y = -2x^{3} - 6x^{2} + 18x + 1$ is increasing or decreasing at $x = -4$.

   Show the worked solution

   1. Step 1: Differentiate$\dfrac{dy}{dx} = -2(3)x^{2} - 6(2)x + 18(1)$$= -6x^{2} - 12x + 18$
   2. Step 2: Put the given x into dy/dx$\dfrac{dy}{dx} = -6(-4)^{2} - 12(-4) + 18$$= -30$
   3. Step 3: Increasing or decreasing? Use the sign of dy/dx$\dfrac{dy}{dx} < 0$, so $y$ is decreasing at $x = -4$.
3. Question 3Where a curve is increasing

   Find the range of values of $x$ for which $y = 2x^{3} - 6x - 2$ is increasing.

   Show the worked solution

   1. Step 1: Differentiate$\dfrac{dy}{dx} = 2(3)x^{2} - 6(1)$$= 6x^{2} - 6$
   2. Step 2: Increasing: is dy/dx greater than or less than 0?$6x^{2} - 6 > 0$
   3. Step 3: Divide by the common factor. Does the inequality sign change?$x^{2} - 1 > 0$
   4. Step 4: Factorise$(x + 1)(x - 1) 0$
   5. Step 5: Sketch the ∪-shaped curve of the left-hand side: which values of x make the inequality true?The curve is above the x-axis outside the roots.
   6. Step 6: Write the range of values of x$x < -1 \text{ or } x > 1$
4. Question 4Where a curve is increasing

   Find the range of values of $x$ for which $y = -x^{3} + 3x^{2} + 9x - 2$ is increasing.

   Show the worked solution

   1. Step 1: Differentiate$\dfrac{dy}{dx} = -3x^{2} + 3(2)x + 9(1)$$= -3x^{2} + 6x + 9$
   2. Step 2: Increasing: is dy/dx greater than or less than 0?$-3x^{2} + 6x + 9 > 0$
   3. Step 3: Divide by the common factor. Does the inequality sign change?$x^{2} - 2x - 3 < 0$
   4. Step 4: Factorise$(x + 1)(x - 3) < 0$
   5. Step 5: Sketch the ∪-shaped curve of the left-hand side: which values of x make the inequality true?The curve is below the x-axis between the roots.
   6. Step 6: Write the range of values of x$-1 < x < 3$
5. Question 5Where a curve is decreasing

   Find the range of values of $x$ for which $y = 2x^{3} - 18x^{2} + 30x + 3$ is decreasing.

   Show the worked solution

   1. Step 1: Differentiate$\dfrac{dy}{dx} = 2(3)x^{2} - 18(2)x + 30(1)$$= 6x^{2} - 36x + 30$
   2. Step 2: Decreasing: is dy/dx greater than or less than 0?$6x^{2} - 36x + 30 < 0$
   3. Step 3: Divide by the common factor. Does the inequality sign change?$x^{2} - 6x + 5 < 0$
   4. Step 4: Factorise$(x - 1)(x - 5) < 0$
   5. Step 5: Sketch the ∪-shaped curve of the left-hand side: which values of x make the inequality true?The curve is below the x-axis between the roots.
   6. Step 6: Write the range of values of x$1 < x < 5$
6. Question 6Where a curve is decreasing

   Find the range of values of $x$ for which $y = 2x^{3} + 6x^{2} - 18x + 1$ is decreasing.

   Show the worked solution

   1. Step 1: Differentiate$\dfrac{dy}{dx} = 2(3)x^{2} + 6(2)x - 18(1)$$= 6x^{2} + 12x - 18$
   2. Step 2: Decreasing: is dy/dx greater than or less than 0?$6x^{2} + 12x - 18 < 0$
   3. Step 3: Divide by the common factor. Does the inequality sign change?$x^{2} + 2x - 3 < 0$
   4. Step 4: Factorise$(x + 3)(x - 1) < 0$
   5. Step 5: Sketch the ∪-shaped curve of the left-hand side: which values of x make the inequality true?The curve is below the x-axis between the roots.
   6. Step 6: Write the range of values of x$-3 < x < 1$