Question

Let \(f,g,h\) be differentiable functions such that \[ \begin{vmatrix} f(x) & g(x) & h(x)\\ f'(x) & g'(x) & h'(x)\\ f''(x) & g''(x) & h''(x) \end{vmatrix} =0. \] Then which of the following statements is correct?

• A. \(f,g,h\) are always linearly independent
• B. \(f,g,h\) are linearly dependent
• C. Exactly two of them are equal
• D. None of these

Hint:

For questions involving determinants of functions and their derivatives, immediately think of the Wronskian. A zero Wronskian is a strong indicator of linear dependence.

Answer 1

The Correct Option is B

Step 1: Name the determinant.
The given determinant, built from the functions and their first and second derivatives, is called the Wronskian. It is a standard tool to test whether functions are linked.

Step 2: Recall what dependence means.
Functions $f,g,h$ are linearly dependent if we can find constants $A,B,C$, not all zero, with $Af(x)+Bg(x)+Ch(x)=0$ for every $x$. This means one function is a blend of the others.

Step 3: Differentiate that relation.
If $Af+Bg+Ch=0$ holds for all $x$, differentiating gives $Af'+Bg'+Ch'=0$, and once more $Af''+Bg''+Ch''=0$.

Step 4: See this as a system.
These three equations form a system whose unknowns are $A,B,C$. The coefficient matrix is exactly the Wronskian matrix shown in the question.

Step 5: Use the zero determinant.
We are told this determinant equals $0$. A homogeneous system with zero determinant has a non-trivial solution, so $A,B,C$ exist that are not all zero.

Step 6: Interpret the result.
Finding such constants is exactly the definition of linear dependence. So the functions are tied together.

Step 7: Conclude.
Under the usual conditions, a vanishing Wronskian signals that the functions are linearly dependent.
\[ \boxed{f,g,h\ \text{are linearly dependent}} \]