Let's do some calculus! (11)

Calculus Level 4

Let $f_{r}(x), g_{r}(x), h_{r}(x)$ be polynomials defined for $r \in \{1,2,3\}$ such that $f_{r}(a)=g_{r}(a)=h_{r}(a)$ for each and every $r \in \{1,2,3\}$ .

Let another polynomial $F(x)$ be defined as:

$F(x) = \begin{vmatrix} f_{1}(x) & f_{2}(x) & f_{3}(x) \\ g_{1}(x) & g_{2}(x) & g_{3}(x) \\ h_{1}(x) & h_{2}(x) & h_{3}(x) \end{vmatrix}$

Find $F'(a)$ .

Notations: $F'(x)$ denotes the first derivative of $F(x)$ .

Clarification:

Each and every $r \in \{1,2,3\}$ means that for any $r$ chosen from the set $\{1,2,3\}$ , $f_{r}(a)=g_{r}(a)=h_{r}(a)$ . Such as, $f_{1}(a)=g_{1}(a)=h_{1}(a)$ , $f_{2}(a)=g_{2}(a)=h_{2}(a)$ and $f_{3}(a)=g_{3}(a)=h_{3}(a)$ respectively.

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The answer is 0.

1 solution

Aditya Narayan Sharma
Sep 21, 2016

Since , $F(x) = \begin{vmatrix} f_{1}(x) & f_{2}(x) & f_{3}(x) \\ g_{1}(x) & g_{2}(x) & g_{3}(x) \\ h_{1}(x) & h_{2}(x) & h_{3}(x) \end{vmatrix}$ then $\displaystyle F'(x) = \begin{vmatrix} f_{1}'(x) & f_{2}'(x) & f_{3}'(x) \\ g_{1}(x) & g_{2}(x) & g_{3}(x) \\ h_{1}(x) & h_{2}(x) & h_{3}(x) \end{vmatrix}+\begin{vmatrix} f_{1}(x) & f_{2}(x) & f_{3}(x) \\ g_{1}'(x) & g_{2}'(x) & g_{3}'(x) \\ h_{1}(x) & h_{2}(x) & h_{3}(x) \end{vmatrix}+\begin{vmatrix} f_{1}(x) & f_{2}(x) & f_{3}(x) \\ g_{1}(x) & g_{2}(x) & g_{3}(x) \\ h_{1}'(x) & h_{2}'(x) & h_{3}'(x) \end{vmatrix}$

and putting $x=a$ reveals that two rows are identical in each of the determinants so the answer is $\boxed{0}$

Let's do some calculus! (11)

For more problems on calculus, click here .

The answer is 0.

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