Is #f(x)=3x^5-x^3+8x^2-x# concave or convex at #x=0#? Calculus Graphing with the Second Derivative Analyzing Concavity of a Function 1 Answer VNVDVI Apr 11, 2018 #f(x)# is concave at #x=0.# Explanation: #f(x)# is concave at #x=a# if #f''(a)>0#. #f(x)# is convex at #x=a# if #f''(a)>0#. Here, #f(x)=3x^5-x^3+8x^2-x, a=0.# Knowing that, let's take the second derivative. #f'(x)=15x^4-3x^2+16x-1# #f''(x)=20x^3-6x+16# Evaluate at #0:# #f''(0)=20(0^3)-6(0)+16=16>0# #f(x)# is concave at #x=0.# Answer link Related questions How do you determine the concavity of a quadratic function? How do you find the concavity of a rational function? What is the concavity of a linear function? What x values is the function concave down if #f(x) = 15x^(2/3) + 5x#? How do you know concavity inflection points, and local min/max for #f(x) = 2x^3 + 3x^2 - 432x#? How do you determine the concavity for #f(x) = x^4 − 32x^2 + 6#? How do you find the intervals on which the graph of #f(x)=5sqrtx-1# is concave up or is concave... How do you determine where the given function #f(x) = (x+3)^(2/3) - 6# is concave up and where... How do you determine the intervals on which function is concave up/down & find points of... On what intervals the following equation is concave up, concave down and where it's inflection... See all questions in Analyzing Concavity of a Function Impact of this question 1169 views around the world You can reuse this answer Creative Commons License