A curve has equation y = 32x² + 1/(8x²) where x ≠ 0.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [B2] dy/dx = 64x - 2x⁻³ oe, isw
2. [B1] for dy/dx = 64x + kx⁻³ or dy/dx = kx - 2x⁻³/8 where k is a non-zero constant
3. [SC1] for dy/dx = 64x - 2x⁻³/8 + c
4. [M1] their dy/dx = 0 and attempt to solve FT their derivative providing it has two terms and at least one term is a correct power of x
5. [A2] (0.25, 4), (-0.25, 4) nfww, isw
6. [A1] for either stationary point correct or for x = ± 0.25 nfww
7. [SC2] if dy/dx = 64x-16x⁻³, then award (± 1/√2, 65/4) oe
8. [SC1] if dy/dx = 64x-16x⁻³, then award for either of these stationary points or x = ± 1/√2 oe
9. [M1] Correct second derivative: d²y/dx² = 64 + 3x⁻⁴ oe, isw FT their dy/dx = mx + nx⁻³ where m ≠ 0 and n ≠ 0 seen in part (a)
10. [A2] dep on x = ±0.25 nfww in part (a)
11. [A1] dep on x = 0.25 or x = -0.25 nfww in part (a) for correctly showing or stating d²y/dx² = 64 + 3/(4x⁴) is positive and minimum [points] oe
12. [A1] OR when x = -0.25, d²y/dx² = 256 or d²y/dx² > 0 oe and minimum [points] oe
13. [A1] OR d²y/dx² = 64 + 3/(4x⁴) and this is positive for any value of x and minimum [points]