Let $-1 \leq p \leq 1$. The equation $4 x^{3}-3 x-p=0$ has in the interval $\left[\frac{1}{2}, 1\ri

Question

Let $-1 \leq p \leq 1$. The equation $4 x^{3}-3 x-p=0$ has  in the interval $\left[\frac{1}{2}, 1\right]$

Professor PC Thomas Classes · Answer

Let $f(x)=4 x^{3}-3 x-p=0$\ we have $f^{\prime}(x)=12 x^{2}-3$ $$ =12\left(x-\frac{1}{2} ight)\left(x+\frac{1}{2} ight) $$ For $\frac{1}{2}0, x+\frac{1}{2}>0$\ This $f^{\prime}(x)>0$, for $\frac{1}{2}