Let a `in` R and f : `R rarr R` be given by `f(x)=x^(5)-5x+a`, then
(a) `f(x)=0` has three real roots if `a gt 4`
(b) `f(x)=0` has only one real root if `a gt 4`
(c) `f(x)=0` has three real roots if `a lt -4`
(d) `f(x)=0` has three real roots if `-4 lt a lt 4`

Explanation

To solve the problem, we need to analyze the function \( f(x) = x^5 - 5x + a \) and determine the conditions under which it has a certain number of real roots based on the value of \( a \).

### Step 1: Find the first derivative
We start by finding the first derivative of the function to identify critical points.
\[ f'(x) = 5x^4 - 5 \]
Setting the derivative equal to zero to find critical points:
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