If f(x) = x^3 + ax^2 + bx + c has a maximum at x = -1 and minimum… - Vidyadip

Question

If $f(x) = x^3 + ax^2 + bx + c$ has a maximum at $x = -1$ and minimum at $x = 3.$ Determine $a, b$ and $c.$

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Answer

We have, $f(x) = x^3 + ax^2+ bx + c$
$\Rightarrow f'(x) = 3x^2+ 2ax + b$
As, $f(x)$ is maximum at $x = -1$ and minimum at $x = 3.$
So, $f(-1) = 0$ and $f(3) = 0$
$\Rightarrow 3(-1)^2 + 2a(-1) + b = 0$ and $3(3)^2 + 2a(3) + b = 0$
$\Rightarrow 3 - 2a + b = 0 ...(i)$
and $27 + 6a + b = 0 ...(ii)$
$(ii) - (i),$ We get
$27 - 3 + 6a + 2a = 0$
$\Rightarrow 8a = -24$
$\Rightarrow a = -3$
Substituting $a = -3$ in $(i)$, we get
$3 - 2(-3) + b = 0$
$\Rightarrow 3 + 6 + b = 0$
$\Rightarrow b = -9$
And, $\text{C}\in R$

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