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