Final answer:
To find the values of x for which the function h(x) = (1-4x^2)/x is decreasing, we need to set the derivative of h(x) less than zero. The intervals where h(x) is decreasing are x < -1/2 and x > 1/2. To determine when the graph of h(x) is concave downward, we need to find the intervals where the second derivative of h(x) is negative. The intervals where h(x) is concave downward are x < -1/2 and x > 1/2. Finally, to find if there is a point of inflection for h(x), we need to determine the values of x where the second derivative changes sign. The second derivative does not change sign for any value of x, so there is no point of inflection for h. Hence the correct answer is option C
Step-by-step explanation:
a) For what values of x is h decreasing?
To determine when the function h(x) = (1-4x^2)/x is decreasing, we need to find the intervals where the derivative of h(x) is negative. Taking the derivative of h(x) with respect to x, we get h'(x) = (-8x^2 + 1)/x^2. Setting h'(x) < 0, we have (-8x^2 + 1)/x^2 < 0. Solving this inequality, we find that x < -1/2 and x > 1/2.
b) For what values of x is the graph of h concave downward?
To determine when the graph of h(x) is concave downward, we need to find the intervals where the second derivative of h(x) is negative. Taking the second derivative of h(x) with respect to x, we get h''(x) = (-24x^2 + 2)/x^3. Setting h''(x) < 0, we have (-24x^2 + 2)/x^3 < 0. Solving this inequality, we find that x < -1/2 and x > 1/2.
c) Determine, if any, point of inflection for the function h.
To determine if there is a point of inflection for the function h(x), we need to find the values of x where the second derivative changes sign. Taking the second derivative of h(x), we have h''(x) = (-24x^2 + 2)/x^3. The second derivative does not change sign for any value of x. Therefore, there is no point of inflection for the function h.
Hence the correct answer is option C