piecewise. Thus, s(5) − s(2) = 568 − 49 = 519.
- Let f be the function given by f(x) = −2x^4 + 6x^2 + 2.
(a) Find the equation for the line normal to the graph at (1, 6).
The line normal to the graph will have a slope that is the opposite reciprocal of the tangent line
at that point. Therefore, begin by finding the slope of the tangent line, i.e., the first derivative. f′
(x) = −8x^3 + 12x and f′ (1) 4. The slope of the normal line is − . So, the equation of the normal
line is: y − 6 = − (x − 1).
(b) Find the x- and y-coordinates of the relative maximum and minimum points.
The relative maximum and minimum will occur at the points when the first derivative is zero or
undefined. In this case, set the first derivative to zero and solve for x: f′(x) = −8x^3 + 12x = 0,
and x = 0, x = , and x = − . To determine which of these is a relative maximum and
which is a relative minimum, find the second derivative at each of these critical points. f′′ (x) =
− 24x^2 + 12, and f′′ (0) = 12, f′′ = − 24, and f′′ = −24. By the second
derivative test, x = 0 corresponds with a relative minimum because f′′(0) > 0 and x = , and
x = − correspond with relative maximums because f′′ and f′′ < 0. To
determine the y-coordinates of these points, plug them back into f(x): f (0) = 2, f =
and f = . So, a relative minimum occurs at (0, 2) and relative maximums occur at
and .
(c) Find the x- and y-coordinates of the points of inflection.
Points of inflection occur when the second derivative equals zero. Take the second derivative
