2019_Question1_a

2019_Question1_a

This is a positive cubic, so we should think about studying its turning points to see what sort of wiggly-cubic-shape graph it gives. We see that \frac{\mathrm{d}}{\mathrm{d}x}(x^3-300x-3000)=3x^2-300, which has roots at \pm10. But both 10^3-300\cdot10-3000 and (-10)^3-300\cdot(-10)-3000 are negative, which means that the local maximum is definitely below the x-axis. We know it must have exactly one real solution then, since complex solutions always come in conjugate pairs. Thus the answer is (b).

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