# Functions and more functions

Calculus Level pending

Let us define 2 functions $$f(x)$$ and $$g(x)$$ such that

$$f ^{\prime \prime} (x) = - f(x)$$ and,

$$g(x) = f^{ \prime }(x)$$

Now lets define a new function $$H(x)$$ such that

$$H(x) = \left( f \left(\dfrac x2\right) \right)^2 + \left( g\left(\dfrac x2\right) \right)^2$$

It is given that $$H(e) = \pi$$. Then find value of $$H(\pi)$$ upto 2 decimal places

Notation:

• $$f ^{\prime} (x)$$ refers to $$\displaystyle \frac{df(x)}{dx}$$.

• $$f ^{\prime \prime} (x)$$ refers to $$\displaystyle \frac{d^2f(x)}{dx^2}$$.

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