Final answer:
The function f(x) = 4x^2 + 3/(x^2 - 4) has no real x-intercept or y-intercept. To find where it's increasing or decreasing, and its concavity, one must calculate the first and second derivatives respectively.
Step-by-step explanation:
To find the x-intercept of the function f(x) = 4x^2 + \frac{3}{x^2 - 4}, we need to set the output, f(x), to zero and solve for x. However, it's clear that no real value of x will satisfy the equation f(x) = 0 since the numerator will always be positive. As for the y-intercept, we find it by substituting x = 0 into the function, which is not possible in this case since the denominator becomes zero. This means the function has no y-intercept either.
To determine where the function is increasing or decreasing, we compute the first derivative f'(x). Using the quotient rule, the derivative is complex but this process will show us where the function is increasing (where f'(x) > 0) and decreasing (where f'(x) < 0).
The concavity of the function can be determined by taking the second derivative, f''(x). The function is concave up where f''(x) > 0 and concave down where f''(x) < 0. Inflection points occur where the concavity changes which is found where the second derivative equals zero or is undefined.