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Answer :
Final answer:
The points on the curve where the tangent line has the largest slope are (−√18, 67) and (√18, 67).
Explanation:
To determine the points on the curve where the tangent line has the largest slope, we need to find the maximum value of the derivative of the curve function.
Given the curve equation y = 1 + 60x³ − 2x⁵, we can find the derivative by taking the derivative of each term:
y' = 0 + 180x² − 10x⁴ = 180x² − 10x⁴
To find the points where the tangent line has the largest slope, we need to find the values of x that make the derivative equal to zero:
180x² − 10x⁴ = 0
Factoring out x²: x²(180 − 10x²) = 0
Setting each factor equal to zero:
x² = 0, x = 0
180 - 10x² = 0
Solving for x: x = ±√(18)
Therefore, the points on the curve where the tangent line has the largest slope are (−√18, 67) and (√18, 67).
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