Analyze and sketch a graph of the function Find any intercep

Solution

We have given the function as f(x) = 12x^(2/3) - 8x we can also represent it as y = 12x^(2/3) - 8x now, to find the y intercept we need to find the slope. We know that derivative of the given function gives the slope. y = 12x^(2/3) - 8x y\' = 8(x)^(1/3) - 8 equating derivative with = 0, will give the values of x slope = y\' = 8(x)^(1/3) - 8 = 0 = 8(x)^(1/3) - 8 x = 1. now, we know from the slope intercept form that y = mx + b where b is the y intercept. plugging x = 1 into the given equation will give y = 12x^(2/3) - 8x y = 4 so, the b value will be y = mx + b 4 = -8*1 + b b = 12 we know that plugging the value of x = 0 will give the y intercept. so y = -8x + b 12 = -8x + b b = 12----------> y intercept so, the slope intercept will be y = -8x + 12 for x intercept y = 0 y = -8x + 12 0 = -8x + 12 x = 12/8 = 3/2--------------> x intercept we will get local maxima at x = 1 Hope this will help you!