Question

The equation of a curve is


`y = 4x {}^(3) + 4px {}^(2) + 16x - 9`
Find the range of values of p such that y is an increasing function.
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Answer 1

Let's solve for p.

`y=4x3+4px2+16x−9`

Step 1: Flip the equation.

`4px2+4x3+16x−9=y`

Step 2: Add -4x^3 to both sides.

`4px2+4x3+16x−9+−4x3=y+−4x3`

`4px2+16x−9=−4x3+y`

Step 3: Add -16x to both sides.

`4px2+16x−9+−16x=−4x3+y+−16x`

`4px2−9=−4x3−16x+y`

Step 4: Add 9 to both sides.

`4px2−9+9=−4x3−16x+y+9`

`4px2=−4x3−16x+y+9`

Step 5: Divide both sides by 4x^2.

4px² / 4x² = −4x³ − 16x + y + 9 / 4x²

Answer:

p = −4x³ − 16x + y + 9 / 4x²