Question

Let $ f(x) $ be twice differentiable such that $ f(x)=-f(x),f(x)=g(x), $ where $ f(x) $ and $ f(x) $ represent the first and second derivatives of $ f(x) $ respectively. Also, if $ h(x)={{[f(x)]}^{2}}+{{[g(x)]}^{2}} $ and $ h(5)=5, $ then $ h(10) $ is equal to:

• A. 3
• B. 10
• C. 13
• D. 5
• E. 0

Answer 1

Answer: (D)

We have, $ h(x)={{\{f(x)\}}^{2}}+{{\{g(x)\}}^{2}} $ On differentiating w.r.t. $ x, $ we get $ \Rightarrow $ $ h(x)=2f(x)f(x)+2g(x)g(x) $ ??(i) Now, $ f(x)=g(x) $ and $ f\,(x)=-f(x) $ $ \Rightarrow $ $ f\,(x)=g(x) $ and $ f\,(x)=-f(x) $ $ \Rightarrow $ $ -f(x)=g(x) $ Thus, $ f(x)=g(x) $ and $ g(x)=-f(x) $ From (i) $ h(x)=-2g(x)g(x)+2g(x)g(x) $ $ =0 $ $ \Rightarrow $ $ h(x)=5 $ $ [\because h(5)=5] $ $ \therefore $ $ h(10)=5 $