Question

If \(f:[-2 a, 2 a] \rightarrow R\) is an odd function such that \(f(x)=f(2 a-x)\) for \(x \in(a, 2 a)\). if the left hand derivative of \(f(x)\) at \(x=a\) is zero, then the left hand derivative of \(f(x)\) at \(x=-a\) is (A) 1 (B) \(-1\) (C) 0 (T)) none

Short answer

Answer: The left-hand derivative of \(f(x)\) at \(x=-a\) is 0.

Step-by-step solution

Recall properties of odd function

An odd function is a function that satisfies the property \(f(-x) = -f(x)\) for all values of \(x\) in its domain.

Evaluate the given function for the specific domain

As the function is defined for \(x \in (a, 2a)\), we can find the relationship between \(f(a-x)\) and \(f(x)\) using the given condition \(f(x) = f(2a-x)\). Let's first replace \(x\) with \(a-x\): \(f(a - x) = f(2a - (a-x))\) This simplifies to: \(f(a - x) = f(a + x)\) Now we know the relationship between \(f(a - x)\) and \(f(a+x)\) for the given domain.

Apply the definition of left-hand derivative at \(x=-a\)

To find the left-hand derivative of \(f(x)\) at \(x=-a\) we need to evaluate: \(\lim_{h \to 0^{-}} \frac{f(-a + h) - f(-a)}{h}\) Using the relationship from Step 2: \(f(a - x) = f(a + x)\), we can write \(f(-a + h)\) as \(f(a - h)\), thus our left-hand derivative becomes: \(\lim_{h \to 0^{-}} \frac{f(a - h) - f(-a)}{h}\)

Use the properties of odd function to find the left-hand derivative at \(x=-a\)

Since \(f(x)\) is an odd function, we can replace \(f(-a)\) with \(-f(a)\). Now our limit is: \(\lim_{h \to 0^{-}} \frac{f(a - h) + f(a)}{h}\) As the left-hand derivative of \(f(x)\) at \(x=a\) is zero, we have: \(\lim_{h \to 0^{-}} \frac{f(a - h) - f(a)}{-h} = 0\) Now, multiplying both the numerator and the denominator of our limit with \(-1\), we get: \(\lim_{h \to 0^{-}} \frac{f(a) - f(a - h)}{h} = 0\) Comparing with the left-hand derivative at \(x=-a\), we find that they are equal: \(\lim_{h \to 0^{-}} \frac{f(a - h) + f(a)}{h} = \lim_{h \to 0^{-}} \frac{f(a) - f(a - h)}{h} = 0\) Hence, the left-hand derivative of \(f(x)\) at \(x=-a\) is 0 which is option (C).