Question

Number of integral values of $a$ for which the function $f(x)=\left(\frac{4 a-7}{3}\right) x^{3}+(a-3) x^{2}+x+5$ is monotonic for every $x \in R$, is

Answer 1

We have $f(x)=\left(\frac{4 a-7}{3}\right) x^{3}+(a-3) x^{2}+x+5$
$\Rightarrow f'(x)=(4 a-7) x^{2}+2(a-3) x+1$
Now, for $f(x)$ to be monotonic,
$f'(x) \geq 0 \forall x \in R$
or $f'(x) \leq 0 \forall x \in R .$
$\Rightarrow D \leq 0 \Rightarrow(a-3)^{2}-(4 a-7) \leq 0$
$\Rightarrow a^{2}-10 a+16 \leq 0$
$\Rightarrow(a-2)(a-8) \leq 0 \Rightarrow a \in[2,8]$
Also, $4 a-7 \neq 0 \Rightarrow a \neq \frac{7}{4}$
So, $a \in[2,8]-\left\{\frac{7}{4}\right\}$.
Hence, number of integral values of a equals $7$.