Question

At what points on the curve \[y = 1 + 40{x^3} - 3{x^5}\] does the tangent line have the largest slope?

Short answer

The largest slope is 240 which lies at the points \(\left( { - 2, - 223} \right)\) and \(\left( {2,225} \right)\).

Step-by-step solution

Determine the slope of the tangent line

Obtain the derivative of the function as shown below:

\(\begin{aligned}{c}y' = \frac{d}{{dx}}\left( {1 + 40{x^3} - 3{x^5}} \right)\\ = 0 + 120{x^2} - 15{x^4}\\ = 120{x^2} - 15{x^4}\end{aligned}\)

Therefore, the slope of the tangent line to the curve at \(x = a\) is \(m\left( a \right) = 120{a^2} - 15{a^4}\).

Determine the intervals of increase and decrease and maximum value

Obtain the derivative of\(m\left( a \right)\)as shown below:

\(\begin{aligned}{c}m'\left( a \right) = \frac{d}{{da}}\left( {120{a^2} - 15{a^4}} \right)\\ = 240a - 60{a^3}\\ = - 60a\left( {{a^2} - 4} \right)\\ = - 60a\left( {a + 2} \right)\left( {a - 2} \right)\end{aligned}\)

With\(a < - 2\)and\(0 < a < 2\),\(m'\left( a \right) > 0\)on the interval\(\left( {0,2} \right)\)and\(\left( {2,\infty } \right)\). For\(a > 2\)and\( - 2 < a < 0\), \(m'\left( a \right) < 0\)on the interval\[\left( { - \infty , - 2} \right)\]and\(\left( { - 2,0} \right)\).

Therefore, the function\(m\)is increasing on the interval\(\left( {0,2} \right)\)and\(\left( {2,\infty } \right)\).\(m\)is decreasing on the interval\[\left( { - \infty , - 2} \right)\]and\(\left( { - 2,0} \right)\). Certainly,\(m\left( a \right) \to - \infty \)as\(a \to \pm \infty \), therefore, the maximum value of\(m\left( a \right)\)should be one of the two local maxima,\(m\left( { - 2} \right)\,\,{\mathop{\rm or}\nolimits} \,\,m\left( 2 \right)\).

What points does the tangent line have the largest slope

However, both the values of\(m\left( { - 2} \right)\)and\(m\left( 2 \right)\)equals as shown below:

\(\begin{aligned}{c}m\left( { \pm 2} \right) = 120 \cdot {2^3} - 15 \cdot {2^4}\\ = 480 - 240\\ = 240\end{aligned}\)

Therefore, the largest slope is 240 which lies at the points \(\left( { - 2, - 223} \right)\) and \(\left( {2,225} \right)\).