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?
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?
Show 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.
Updated On: Jun 10, 2026
- \(f,g,h\) are always linearly independent
- \(f,g,h\) are linearly dependent
- Exactly two of them are equal
- None of these
Show Solution
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The Correct Option is B
Solution and Explanation
Concept:
The determinant
\[
\begin{vmatrix}
f & g & h\\
f' & g' & h'\\
f'' & g'' & h''
\end{vmatrix}
\]
is called the Wronskian of the functions \(f,g,h\).
A vanishing Wronskian is one of the standard tests used to identify linear dependence among differentiable functions under appropriate conditions.
Step 1: Recall the definition of linear dependence Functions \(f,g,h\) are linearly dependent if there exist constants \(A,B,C\), not all zero, such that \[ Af(x)+Bg(x)+Ch(x)=0 \] for all \(x\) in the interval.
Step 2: Interpret the determinant The determinant given is \[ W(f,g,h)=0 \] where \(W\) denotes the Wronskian. A Wronskian identically equal to zero indicates that the functions fail to generate three independent directions in the function space. Thus they satisfy a non-trivial linear relation.
Step 3: Conclude Hence \[ f,g,h \] must be linearly dependent. Therefore, \[ \boxed{\text{\(f,g,h\) are linearly dependent}} \]
Step 1: Recall the definition of linear dependence Functions \(f,g,h\) are linearly dependent if there exist constants \(A,B,C\), not all zero, such that \[ Af(x)+Bg(x)+Ch(x)=0 \] for all \(x\) in the interval.
Step 2: Interpret the determinant The determinant given is \[ W(f,g,h)=0 \] where \(W\) denotes the Wronskian. A Wronskian identically equal to zero indicates that the functions fail to generate three independent directions in the function space. Thus they satisfy a non-trivial linear relation.
Step 3: Conclude Hence \[ f,g,h \] must be linearly dependent. Therefore, \[ \boxed{\text{\(f,g,h\) are linearly dependent}} \]
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