91Ó°ÊÓ

Find the points \((c, f(c))\) where the line tangent to the graph of \(f(x)=x^{3}-x\) is parallel to the secant line that passes through the points \((-1, f(-1))\) and \((2, f(2))\).

Find the numbers \(x\) for which (a) \(f^{\prime \prime}(x)=0\), (b) \(f^{\prime \prime}(x)>0,\) (c) \(f^{\prime \prime}(x)<0\). $$f(x)=x^{3}$$

Show that the sum of the \(x\) - and \(y\) -intercepts of any line tangent to the graph of \(x^{1 / 2}+y^{1 / 2}=c^{-2}\) is constant and equal to \(c\).

Find a function \(y=f(x)\) with the given derivative. Check your answer by differentiation. $$\frac{d y}{d x}=2\left(x^{3}-2\right)\left(3 x^{2}\right)$$

The double-angle formula for the sine function takes the form: \(\sin 2 x=2 \sin x \cos x .\) Differentiate this formula to obtain a double-angle formula for the cosine function.

Find the points \((c, f(c))\) where the line tangent to the graph of \(f(x)=x /(x+1)\) is parallel to the secant line that passes through the points \((1, f(1))\) and \((3, f(3))\).

Find a function \(y=f(x)\) with the given derivative. Check your answer by differentiation. $$\frac{d y}{d x}=3 x^{2}\left(x^{3}+2\right)^{2}$$

Find the numbers \(x\) for which (a) \(f^{\prime \prime}(x)=0\), (b) \(f^{\prime \prime}(x)>0,\) (c) \(f^{\prime \prime}(x)<0\). $$f(x)=x^{4}$$

Find the numbers \(x\) for which (a) \(f^{\prime \prime}(x)=0\), (b) \(f^{\prime \prime}(x)>0,\) (c) \(f^{\prime \prime}(x)<0\). $$f(x)=x^{4}+2 x^{3}-12 x^{2}$$

Let \(f(x)=1 / x, x>0 .\) Show that the triangle that is formed by each line tangent to the graph of \(f\) and the coordinate axes has an area of 2 square units.