minimum and maximum value

Question

Find the maximum and minimum value of the function 6x⁵-10x²+14x⁴

Answer 1

Find all maxima, minima ...

Step 1 ... find the derivative dy/dx= 30x^4-20x+64x^3

Step 2 ... find all x's where dy/dx =0 ... x=0, -.503, -.676, -1.96 ... these are the critical values

Step 3 ... find the second derivative d2y/dx2 = 120x^3+192x^2-20

Step 4 ... evaluate the second derivative at all critical values (0,-20), (-.503, 12.8), (-.676, -51.6), (-1.96, -215.8)

Step 5 ... identify local values of y at critical values as Maxima if second derviative <0 ... (0,-.676, -1.96)

Step 6 ... identify local values of y at critical values as Minima if second derivative > 0 .. (-.503)

Step 7 ... identify absolute Maxima as + infinity as y approaches infinity as x approaches infinity

Step 8 ... identify absolute Minima as - infinity as y approaches - infinity as x approaches - infinity